Automated Generation of Transit Maps
Dissertation zur Erlangung des Doktorgrades der
Ingenieurwissenschaften der Technischen Fakultät der
Albert-Ludwigs-Universität Freiburg
vorgelegt von
Patrick Brosi
Albert-Ludwigs-Universität Freiburg
Technische Fakultät
Institut für Informatik
2022
Dekan
Prof. Dr. Roland Zengerle
Erstgutachterin und Betreuerin der Arbeit
Prof. Dr. Hannah Bast
Zweitgutachter
J.-Prof. Dr. Thomas van Dijk
Datum der mündlichen Prüfung
10. August 2022
i
Abstract
We consider the problem of automatically drawing public transit maps from schedule
data. The maps may either be geographically accurate (with lines closely following
their real-world course), or schematic. In particular, we study the following subprob-
lems: Given schedule data without geographical line courses (shapes), how can these
shapes be extracted from geographical datasets? Given then a collection of such shaped
vehicle trips, how can we construct a topological network graph whose edge segments
will not overlap in a rendering, and are labeled with the set of lines passing through
them? How can we eciently nd line label permutations that will produce as few line
crossings and/or separations as possible in the nal map? How can we eciently draw a
schematic version of the map which may also consider obstacles, or respect the original
geographical line courses to some degree?
We rst describe a map-matching approach for schedule data based on a hidden
Markov model and perform a quality evaluation on several schedule datasets. In this
context, we also study various methods for deciding whether two public transit station
identiers are similar. Several speed-up techniques for the map-matching process are
additionally presented and evaluated.
We then proceed to describe and evaluate a tailored map construction approach to
build the topological network graph from such data. This approach considers line turn
restrictions and clusters similar stations into single nodes.
To nd optimal line label permutations, we formulate a variant of the classic Metro
Line Crossing Minimization Problem (MLCM) better suited for rendering, which we call
the Metro Line Node Crossing Minimization Problem (MLNCM). We give a weighted
variant (MLNCM-W) and a variant which also considers line separations (MLNCM-WS),
a concept we consider important for the quality of the rendered maps. We prove the
NP-hardness of these problems and describe various approximation methods, as well as
an integer linear program, to solve them to optimality. To speed up solution times and
to improve the approximation quality, we formulate a set of line graph transformation
rules which implicitly compute an optimal partial ordering of the lines and typically
break up the problem into many smaller subproblems.
In the context of schematic maps we describe a novel method for drawing octilinear
transit maps. Both an integer linear program and a fast approximation method based on
the repeated calculation of shortest paths in a special octilinear grid graph are presented.
We evaluate several speed-up techniques and demonstrate that the method also works
for other layouts (e.g. orthoradial or hexalinear).
This work also provides a practical overview of how to nally render the maps in
an esthetically pleasing way. All our approaches were implemented as separate publicly
available tools, forming an easily extendable pipeline.
ii
Zusammenfassung
Wir behandeln das Problem der automatisierten Erstellung von ÖPNV-Karten aus Fahr-
plandaten. Die Karten können dabei sowohl geograsch korrekt, als auch schematisch
sein. Insbesondere untersuchen wir folgende Teilprobleme: Wie können fehlende geo-
grasche Linienverläufe (Shapes) aus vorhandenem Kartenmaterial extrahiert werden?
Wie kann aus einer Menge von Fahrten mit Shapes das darin enthaltene topologische
Netzwerk so abgeleitet werden, dass Kanten mit darauf verkehrenden Linien versehen
sind und sich später nicht überlappen? Wie können schnell Permutationen dieser Linien
gefunden werden, so dass sie sich in der Karte so selten wie möglich kreuzen und/oder
trennen? Wie können ezient schematische Karten erzeugt werden, die auch Hinder-
nisse oder die ursprünglichen geograschen Linienverläufe berücksichtigen?
Wir beschreiben zunächst einen auf einem Hidden-Markov-Modell basierenden Map-
Matching-Ansatz für Fahrplandaten und führen eine Evaluation auf mehreren Fahrplan-
datensätzen durch. Vor diesem Hintergrund untersuchen wir auch Methoden um zu ent-
scheiden ob Paare von Stationen ähnlich sind. Wir untersuchen und evaluieren außer-
dem mehrere Methoden zur Beschleunigung des Map-Matchings.
Weiter beschreiben und evaluieren wir einen maßgeschneiderten Map-Construction-
Ansatz zur Extraktion des toplogischen Netzwerks. Dieser Ansatz kann Abbiegeverbote
von Linien berücksichtigen und ähnliche Stationen in einzelne Knoten überführen.
Um optimale Permutationen der Linien zu nden, formulieren wir eine Variante des
klassischen Metro Line Crossing Minimization Problems (MLCM), die das Rendern der
Karten vereinfacht. Wir nennen diese Variante das Metro Line Node Crossing Minimiza-
tion Problem (MLNCM) und beschreiben außerdem eine gewichtete Variante (MLNCM-
W) sowie eine Variante die auch Linientrennungen berücksichtigt (MLNCM-WS). Letz-
teres erachten wir als sehr wichtig für die Kartenqualität. Wir beweisen die NP-Schwere
dieser Probleme und beschreiben sowohl Näherungsmethoden, als auch ein ganzzahliges
lineares Programm (ILP) zum Finden optimaler Lösungen. Zur schnelleren Optimierung
und zur Qualitätsverbesserung der Näherungsmethoden formulieren wir verschiedene
Transformationsregeln, die implizit optimale partielle Ordnungen auf den Linien berech-
nen und das Problem typischerweise in viele kleinere Unterprobleme teilen.
Für die Schematisierung entwerfen wir eine neue Methode zum Erstellen oktilinearer
Karten. Hier beschreiben wir sowohl ein ILP als auch einen schnellen Approximations-
algorithmus basierend auf der Berechnung kürzester Pfade in einem oktilinearen Git-
tergraphen. Wir untersuchen außerdem Beschleunigungsansätze und zeigen, dass unser
Verfahren auch mit anderen Layouts (z.B. orthoradial oder hexalinear) funktioniert.
In dieser Arbeit wird außerdem ein praxisorientierter Überblick über das ästhetisch
ansprechende Rendern der Karten gegeben. Alle unsere Methoden wurden in separaten
und öentlich verfügbaren Tools implementiert, die zusammen eine einfach zu erwei-
ternde Pipeline bilden.
iii
Acknowledgments
My sincere thanks go to Hannah Bast for her advice and guidance during the 5 years
I worked on this thesis, and for all her thoughtful comments during nightly sessions
before paper deadlines. I am extremely grateful for the opportunities, freedom, and time
provided to me at the Chair of Algorithms and Data Structures. Special thanks go to
Sabine Storandt for all her valuable insights and her support with linear programming.
Both Hannah and Sabine gave many additional ideas that unfortunately did not nd
their way into this thesis due to time constraints.
I would also like to thank my colleagues and former master students at our chair
Björn Buchhold, Thomas Goette, Matthias Hertel, Johannes Kalmbach, Theresa Klumpp,
Claudius Korzen, Axel Lehmann, Markus Näther, Natalie Prange, and Niklas Schnelle for
an always very pleasant working environment, frequent discussions over computer sci-
ence related topics as well as real-world trivia, and cooking sessions in our kitchen. Ad-
ditional thanks go to Natalie, Matthias, and Axel for their thorough proofreading. I also
wish to thank our secretary Heike Hägle for her help in navigating the administrative
jungle, and our administrator Frank Dal-Ri for his technical support.
Some of the anonymous reviewers of the publications preceding this work went out
of their way to provide suggestions for improvement. I would also like to express my
gratitude for these helpful comments.
Finally, I want to thank my family, especially my wife Manuela, for all their support
and love.
iv
Contents
1 Introduction 1
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.1 Schedule Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.2 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.3 Shortest Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.4 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Map-Matching Schedule Data 15
2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.2 A Survey on Shape Availability and Quality . . . . . . . . . . . . 17
2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.1 Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.2 Hidden Markov Model . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3.1 Map-Matching of GPS Traces . . . . . . . . . . . . . . . . . . . . 21
2.3.2 Map-Matching of Schedule Data . . . . . . . . . . . . . . . . . . 24
2.3.3 Routing with Turn Restrictions . . . . . . . . . . . . . . . . . . . 25
2.3.4 Curve Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.5 Similarity Classication of String Labels and POIs . . . . . . . . . 25
2.4 Baseline Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4.1 Node-To-Set Routing . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4.2 Local Set-to-Set Routing . . . . . . . . . . . . . . . . . . . . . . . 26
2.4.3 Global Set-to-Set Routing . . . . . . . . . . . . . . . . . . . . . . 26
2.5 Hidden Markov Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
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vi Contents
2.5.1 Emissions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5.2 Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5.3 Transformation to Log Probabilities . . . . . . . . . . . . . . . . . 32
2.5.4 Optimal Hidden State Sequence as a Shortest Path Problem . . . 32
2.6 Turn Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.6.1 Mandatory Turn Restrictions . . . . . . . . . . . . . . . . . . . . 35
2.6.2 Angular Turn Costs . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.6.3 Inter-Hop Turn Restrictions . . . . . . . . . . . . . . . . . . . . . 35
2.6.4 Edge-to-Node Dual . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.7 Incorporating Network Meta Data . . . . . . . . . . . . . . . . . . . . . . 41
2.7.1 Public Transit Route Relations . . . . . . . . . . . . . . . . . . . . 42
2.7.2 Station Labels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.8 Advanced Station Similarity Classication . . . . . . . . . . . . . . . . . 43
2.8.1 Characteristics of Station Identiers . . . . . . . . . . . . . . . . 43
2.8.2 Classication Techniques . . . . . . . . . . . . . . . . . . . . . . 45
2.9 Map-Matching Speedup Techniques . . . . . . . . . . . . . . . . . . . . . 51
2.9.1 Faster Layer-to-Layer Shortest Path Calculations . . . . . . . . . 52
2.9.2 A
∗
Heuristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.9.3 Early Stopping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.9.4 Bulk Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.10 Experimental Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.10.1 Quality of the Station Similarity Classication Methods . . . . . 59
2.10.2 Quality of Our Map-Matching Approach . . . . . . . . . . . . . . 64
2.10.3 Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.11 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . 72
3 Line Graph Construction 75
3.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.1.1 Map Construction . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.1.2 Quality Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.1.3 Edge Bundling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.2 Support Graph Construction . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.2.1 Artifacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.2.2 Line Creep and Network Planarity Issues . . . . . . . . . . . . . . 83
3.2.3 Convergence Criterion . . . . . . . . . . . . . . . . . . . . . . . . 85
vii
3.2.4 Intersection Smoothing . . . . . . . . . . . . . . . . . . . . . . . 86
3.3 Inferring Line Turn Restrictions . . . . . . . . . . . . . . . . . . . . . . . 86
3.3.1 Edge Handles and Shortest Path Length Comparison . . . . . . . 87
3.3.2 Shortest Path Comparison . . . . . . . . . . . . . . . . . . . . . . 88
3.3.3 Avoiding False Positives . . . . . . . . . . . . . . . . . . . . . . . 90
3.4 Station Clustering and Insertion . . . . . . . . . . . . . . . . . . . . . . . 90
3.5 Experimental Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.5.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.5.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 92
3.6 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . 95
4 Line Ordering Optimization 103
4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.1.1 Crossing Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.1.2 Metro Line Crossing Minimization . . . . . . . . . . . . . . . . . 106
4.1.3 Periphery Condition . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.3 Weighted MLNCM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.4 Minimizing Line Separations . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.5 NP-Hardness Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.6 Heuristic Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.6.1 Ecient Calculation of the Target Function . . . . . . . . . . . . 118
4.6.2 Exhaustive Search . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.6.3 Greedy Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.6.4 Greedy Search with Lookahead . . . . . . . . . . . . . . . . . . . 120
4.6.5 Steepest-Ascent Hill Climbing . . . . . . . . . . . . . . . . . . . . 122
4.6.6 Simulated Annealing . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.7 Baseline ILP Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.7.1 Line Positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.7.2 Counting Line Crossings . . . . . . . . . . . . . . . . . . . . . . . 123
4.7.3 Counting Line Separations . . . . . . . . . . . . . . . . . . . . . . 124
4.7.4 ILP Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.8 Improved ILP Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.8.1 Range Line Positions . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.8.2 Crossing Oracle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
viii Contents
4.8.3 Line Separations . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.8.4 Improved ILP Size . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.9 Line Graph Simplication . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.9.1 General Observations . . . . . . . . . . . . . . . . . . . . . . . . 130
4.9.2 Pruning Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4.9.3 Cutting Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4.9.4 Untangling Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.9.5 Full Simplication . . . . . . . . . . . . . . . . . . . . . . . . . . 145
4.9.6 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
4.10 Full Solve Through Simplication . . . . . . . . . . . . . . . . . . . . . . 157
4.10.1 Tree-Like Line Graphs . . . . . . . . . . . . . . . . . . . . . . . . 157
4.10.2 Two-Sided Tree-Like Line Graphs . . . . . . . . . . . . . . . . . . 159
4.10.3 Real-World Examples . . . . . . . . . . . . . . . . . . . . . . . . . 160
4.11 Experimental Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
4.11.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
4.11.2 Comparison of Heuristic Approaches . . . . . . . . . . . . . . . . 164
4.11.3 Comparison of Linear Programs . . . . . . . . . . . . . . . . . . . 167
4.11.4 Eects of Full Line Graph Simplication . . . . . . . . . . . . . . 169
4.11.5 Comparison to Manually Designed Maps . . . . . . . . . . . . . . 172
4.12 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . 173
5 Schematic Transit Maps 177
5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
5.1.1 Degree-2 Heuristic . . . . . . . . . . . . . . . . . . . . . . . . . . 178
5.1.2 Station Labels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
5.1.3 Extending the Maximum Input Node Degree . . . . . . . . . . . . 179
5.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
5.2.1 Generation of Schematic Maps . . . . . . . . . . . . . . . . . . . 180
5.2.2 Generation of Schematic Transit Maps . . . . . . . . . . . . . . . 181
5.2.3 Generation of Non-Octilinear Schematic Transit Maps . . . . . . 185
5.2.4 General Algorithmic Work . . . . . . . . . . . . . . . . . . . . . . 186
5.3 Schematization on Octilinear Grid Graphs . . . . . . . . . . . . . . . . . 187
5.3.1 Line Bend Penalties . . . . . . . . . . . . . . . . . . . . . . . . . . 187
5.3.2 Modeling Edge Weights in the Extended Grid Graph . . . . . . . 189
5.3.3 Full Optimization Problem . . . . . . . . . . . . . . . . . . . . . . 193
ix
5.3.4 NP-Hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
5.4 Integer Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . 194
5.4.1 Image Nodes and Station Displacement . . . . . . . . . . . . . . . 195
5.4.2 Edge Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
5.4.3 Topology Preservation . . . . . . . . . . . . . . . . . . . . . . . . 196
5.4.4 Line Bend Minimization . . . . . . . . . . . . . . . . . . . . . . . 198
5.4.5 Asymptotic ILP Size . . . . . . . . . . . . . . . . . . . . . . . . . 199
5.5 Approximate Solution via Iterative Shortest Path Calculation . . . . . . . 199
5.5.1 Input Edge Ordering . . . . . . . . . . . . . . . . . . . . . . . . . 200
5.5.2 Edge Routing and Station Placement . . . . . . . . . . . . . . . . 201
5.5.3 Topology Preservation . . . . . . . . . . . . . . . . . . . . . . . . 202
5.5.4 Line Bend Minimization . . . . . . . . . . . . . . . . . . . . . . . 203
5.5.5 Stalling Prevention Heuristics . . . . . . . . . . . . . . . . . . . . 203
5.5.6 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
5.5.7 Local Search Optimization . . . . . . . . . . . . . . . . . . . . . . 206
5.5.8 Distance Optimization Between Contracted Degree-2 Nodes . . . 207
5.6 Speedup Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
5.6.1 Shortest-Path Heuristic . . . . . . . . . . . . . . . . . . . . . . . 208
5.6.2 Sparse Base Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
5.6.3 Iterative Octilinear Hanan Grids . . . . . . . . . . . . . . . . . . 212
5.7 Obstacles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
5.8 Approximating Geographical Courses . . . . . . . . . . . . . . . . . . . . 213
5.9 Hard Constraint Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . 215
5.10 Non-Octilinear Layouts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
5.10.1 Hexalinear Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
5.10.2 Orthoradial Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
5.11 Experimental Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
5.11.1 Integer Linear Program . . . . . . . . . . . . . . . . . . . . . . . 219
5.11.2 Input Edge Orderings for Approximate Approach . . . . . . . . . 221
5.11.3 Quality and Speed of Approximate Approach . . . . . . . . . . . 222
5.11.4 Eect of Sparse Base Grids . . . . . . . . . . . . . . . . . . . . . . 224
5.11.5 Non-Octilinear Layout Experiments . . . . . . . . . . . . . . . . 228
5.11.6 Visual Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
5.12 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . 229
x Contents
6 Rendering and Labeling 241
6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
6.2 Osetted Line Rendering . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
6.3 Inner Node Line Connections . . . . . . . . . . . . . . . . . . . . . . . . 244
6.4 Station Rendering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
6.4.1 Fully Served Stations . . . . . . . . . . . . . . . . . . . . . . . . . 246
6.4.2 Partially Served Stations . . . . . . . . . . . . . . . . . . . . . . . 248
6.5 Labeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
7 Software 251
7.1 pfaedle - Map-Matching GTFS Data . . . . . . . . . . . . . . . . . . . . . 251
7.2 statsimi - Similarity Classication of Public Transit Stations . . . . . . . 252
7.3 staty - Quality Assessment of OSM Station Tagging . . . . . . . . . . . . 253
7.4 topo - Map Construction for Line Graphs . . . . . . . . . . . . . . . . . . 253
7.5 loom - Line-Ordering Optimization . . . . . . . . . . . . . . . . . . . . . 254
7.6 octi - Fast Schematization of Line Graphs . . . . . . . . . . . . . . . . . . 254
7.7 transitmap - Transit Map Rendering Engine . . . . . . . . . . . . . . . . 255
8 Conclusions 257
List of Figures 266
List of Tables 268
Bibliography 269
Chapter 1
Introduction
Transit maps are diagrams of public transportation networks. They depict the lines and
stations of e.g. subway, bus, rail, or tram systems, but may also show points of interest
or geographical landmarks. Their primary function is to give passengers a network
overview to help them plan their route. To this end, transit maps must clearly indicate
which stops are served by which lines. They should also respect the locations of stops
as well as the geographical line course, at least to some extent.
Historically, transit maps had a high degree of geographical accuracy. In 1931, Harry
Beck proposed his famous schematic design of the London subway network [68, 92]
(he was not the rst to use schematic maps [45]). In his design, edge segment ori-
entations were always multiples of 45 degrees (see Figure 1.1). While there are other
schematization methods used in real-world transit maps (already in 1933, a map of the
Berlin S-Bahn system featured an orthoradial-like layout [82]), this so-called octilinear
(or octolinear [151]) design has since become the de facto standard and has largely re-
placed geographically accurate transit maps.
Map services like OpenStreetMap, Google Maps, or Bing Maps, however, require
geographically accurate transit maps which can be used as overlays over existing maps
or satellite imagery. In particular, the global coverage of these map services requires an
automated generation from raw schedule data, as designing and curating them by hand
on a global scale is unrealistic, and schedule data is often the only dataset available. Most
map services already oer such an automatically generated overlay, but the quality is
often disappointing (see Figure 1.2).
For maps used in print, the automated generation of schematic transit maps is also
relevant. Such systems may be used for fast prototyping of network extensions, auto-
mated creation of planning documents (see Figure 1.3), or for the (assisted) creation of
passenger maps. In particular, assistance systems in map editors require fast generation
times. Some transit maps also feature a gradual level of schematization (see for example
Figure 1.4), which an automated approach should also allow for.
1
2 Chapter 1. Introduction
Figure 1.1: Left: 1908 map of the London subway network. Line courses have a high
degree of geographical accuracy and are used as an overlay over a city map. Right:
Octilinear map proposed by Harry Beck in 1931. It is geographically accurate only to a
limited extent. All line segments follow vertical, horizontal, or diagonal segments.
This gives rise to several algorithmic challenges: Given only schedule data (and map
data describing the physical transportation network), how can the geographical line
courses (shapes) be obtained eciently, and in high quality? Given such shapes, how
can we construct the underlying network graph in such a way that overlapping seg-
ments are avoided in the nal map, and enough space around them is left to render the
lines? How can we avoid unnecessary crossings (or separations) between lines in the
nal map? How can we schematize a network fast enough for the method to be used in
map editors? How can the schematization process respect obstacles, and/or follow the
original geographic line courses to some extent?
In this work, we address these challenges and describe a complete and ecient
pipeline to automatically generate such transit maps from raw schedule data. Despite
extensive research in this area over the last two decades, this problem was never ad-
dressed in its entirety. We identify and describe the key algorithmic problems of each
step, briey discuss existing approaches, describe alternative or adapted solutions where
reasonable and evaluate our results on public transit networks from around the world.
All our methods have been implemented in publicly available tools.
1.1 Overview
This work is organized as follows: in the remainder of this introduction, we provide a
high-level overview of our pipeline and contributions (Section 1.2) and give an overview
over concepts, notations and denitions used throughout this work (Section 1.3). All
1.2 Contributions 3
Figure 1.2: Left: The Chicago loop in the transit layer of Google Maps at the publication
time of our work on geographically accurate transit maps. Lines are impossible to follow
visually. Middle: The same map extent on Here maps. Right: The ocial Chicago Transit
Authority map of the same area.
chapters are then preceded with a list of their related work, a detailed list of contribu-
tions, and an overview of the notations and concepts used only in the respective chapter.
In Chapter 2, we describe how the geographical course of a public transit vehicle
can be extracted from geographical data like OpenStreetMap given only the stations
(described by a station identier, namely the coordinates and label) served by the ve-
hicle. In Chapter 3, our goal is to infer the topological network graph (called the line
graph) of a public transit system, given the geographical courses of its vehicles as input.
Chapter 4 presents and evaluates various approaches for nding the optimal line order-
ing on individual segments in the nal drawing. In Chapter 5 we then describe how to
automatically nd schematic drawings of line graphs. Chapter 6 then gives a hands-on
description how the line graphs can be rendered into nal maps. In this context we also
discuss labeling, although we emphasize that further research is required here. Finally,
Chapter 7 gives an overview over the software developed in the course of this work.
1.2 Contributions
We consider the following the main contributions of this work:
We adapt a state-of-the art approach for map-matching to work with very sparse
public transit vehicle trajectories as they typically appear in real-world schedule data. In
this data, trajectories are usually only sampled at stations, with straight-line geometries
in between. A dual-graph representation of the underlying transportation network is
4 Chapter 1. Introduction
Figure 1.3: Excerpt from the ocial 2019 Swiss timetable map which depicts the nation-
wide clock-face schedule. The octilinear map is created manually each year and aimed
at professional transit planers as well as the interested public.
used to include turn restrictions into the map-matching process. We motivate this with
a comprehensive study of the shape quality in public schedule datasets. Several speedup
techniques are also examined. Preliminary results of an earlier version of our approach
have been published in the proceedings of the 26th ACM SIGSPATIAL International
Conference on Advances in Geographic Information Systems (GIS 2018). The results
can be explored via TRAVIC
1
, a real-time visualization tool which is served by a highly
ecient backend. Both TRAVIC and the backend have been published as separate works
in the proceedings of the 22nd ACM SIGSPATIAL International Conference on Advances
in Geographic Information Systems (GIS 2014) [18, 17].
We evaluate and develop methods to check whether two station identiers as they
usually appear in schedules or geographical databases (consisting of a geographical co-
ordinate and a label) are similar. This is a problem that arose multiple times in the context
of this work. Preliminary results have been published in [15].
We describe and evaluate a method that is able to infer the network graph (the line
graph) of a collection of vehicle trajectories. To achieve this, we develop a map construc-
tion technique which is tailored at producing line graphs t for rendering (which we call
free line graphs).
In the context of metro map drawing, we give a formulation of the classic Metro-Line
Crossing Minimization problem (MLCM) which is better suited for rendering (MLNCM)
1
https://travic.app
1.2 Contributions 5
Neckar
Rhein
Donau
Donau
(Baden)
öcherberg
Gengenbach
Biberach (Baden)
Ibach
Zell (Harmersbach)
Birach
Unterharmersbach
Kirnbach-Grün
Oberharmersbach Dorf
Oberharmersbach-Riersbach
Steinach (Baden)
Lahr (Schwarzwald)
St. Georgen
Triberg
Hornberg
Gutach Freilichtmuseum
Oberndorf
(Neckar)
Sulz (Neckar)
Alpirsbach
Schenkenzell
Schiltach Mitte
Schiltach
Halbmeil
Loßburg-Rodt
F
Indus
Mühlen
Bittelbr
Schopfloch
(b Freudenstadt)
Wolfach
Orschweier
Ringsheim
Herbolzheim
Kenzingen
Breisach
Endingen a.K.
Sasbach a.K.
Jechtingen
Burkheim-Bischoffingen
Oberrotweil
Achkarren
Nimburg (Bd)
Hugstetten
Freiburg West
Ihringen
Wasenweiler
Bad Krozingen
Ost
Staufen
Staufen Süd
Etzenbach
Dietzelbach
Münstertal
Freiburg Messe/Universität
Freiburg Klinikum
Freiburg (Brsg) Hbf
Weil am
Rhein
Waldshut
Villingen (Schwarzw)
Hausach
Horb
Ulm Hbf
Trossingen
Bf
Rottweil
Donau-
eschingen
Konstanz
Friedrichshafen (FN) Stadt
Hergatz
Kißlegg
Memming
Tuttlingen
Singen
(Hohentwiel)
Radolfzell
Sigmaringen
Herbertingen
Aulendorf
Schaffhausen
Koblenz
(CH)
Basel Bad Bf
ulhouse
Lörrach Hbf
Gundelfingen (Breisgau)
Freiburg-Zähringen
Freiburg-Herdern
Bahlingen-Riedlen
Bahlingen a. K.
Riegel Ort
Gottenheim
Riegel/Malterdingen
Elzach
Saisonal,
sonst Halt
einzelner Züge
Waldkirch
Kollnau
Gutach (Breisgau)
Bleibach
Niederwinden
Oberwinden
Emmendingen
Kollmarsreute
Buchholz
Batzenhäusle
Kirchzarten
Freiburg-Wiehre
Freiburg-Littenweiler
Freiburg-St. Georgen
Ebringen
Schallstadt
Norsingen
Bad Krozingen
Heitersheim
Buggingen
Müllheim (Baden)
Neuenburg (Baden)
Auggen
Seebrugg
Schluchsee
Aha
Altglashütten-Falkau
Wehr-Brennet
Schwörstadt
Döggingen
Unadingen
Bachheim
Thayngen
Stetten
Hausen i. Tal
Beuron
Fridingen
Mühlheim
Tuttlingen Zentrum
Tuttlingen Nord
Tuttlingen Gansäcker
Nendingen
Rad.-Haselbrunn
Stahringen
Wahlwies
Nenzingen
Stockach
Munderkingen
Herblingen
Neuhausen Bad Bf
Beringen Bad Bf
Beringerfeld
Neunkirch
Wilchingen-Hallau
Neuhausen (CH)
Neuhausen Rheinfall
Jestetten
Lottstetten
Trasadingen
Erzingen (Baden)
Rötenbach (Baden)
Feldberg-Bärental
Grießen (Baden)
Tiengen (Hochrhein)
Lauchringen
Lauchringen West
Wutöschingen
Dogern
Albbruck
Laufenburg Ost
Laufenburg (Baden)
Murg (Baden)
Bad Säckingen
Fahrnau
Hausen-Raitbach
Zell (Wiesental)
Schliengen
Weil-Gartenstadt
Weil Ost
Lörrach Dammstr.
Lörrach
Schwarzwaldstr.
Lörrach-Haagen/Messe
Lörrach-Stetten
Lörrach Museum/Burghof
Weil-Pfädlistraße
Haltingen
Schopfheim
Schopfheim-Schlattholz
Schopfheim West
Maulburg
Steinen
Lörrach-Brombach/Hauingen
Beuggen
Rheinfelden (Baden)
Herten (Baden)
Wyhlen
Grenzach
Eimeldingen
fringen-Kirchen
Istein
Kleinkems
Rheinweiler
Bad Bellingen
Himmelreich
Neustadt
(Schwarzw)
Haslach
Titisee
Böhringen-Rickelsh.
Singen Industriegebiet
Gottmadingen
Bietingen
Immendingen
Immendingen-
Zimmern
Albstadt-Ebingen
Straßberg-Winterlingen
Albstadt-Lautlingen
Albstadt-Laufen Ort
Balingen Süd
Hechingen
Hechingen
Landesbahn
Rangendingen
Storzingen
Bermatingen-Ahausen
Salem
Überlingen Nußdorf
Überlingen
Überlingen Therme
Sipplingen
Ludwigshafen (Bodensee)
Mengen
Sigmaringendorf
Trochtelfingen
Trochtelfingen ALB-GOLD
Haidkapelle
eutlingen
Hbf
Hütten
Sondernach
Münsingen
Gomadingen
Offenhausen
Kohlstetten
Engstingen
Mägerkingen
Gammertingen
Hettingen (Hohenz)
Hermentingen
Veringenstadt
Veringendorf
Jungnau
Hanfertal
Herbertingen Ort
Ehingen
(Donau)
Marbach
Marbach-Grafeneck
ieringen
Eyach
Bad Niedernau
en-Derendingen
Ulm-Söflingen
Blaustein
Herrlingen
Gerhausen
laubeuren
Schelklingen
Ulm O
Dußlingen
Nehren
Mössingen
Bad Sebastiansweiler-Belsen
Bodelshausen
Ulm Donautal
Erbach (Württ)
Schmiechen
Schmiechen Albbahn
Allmendingen
Rottenacker
Laupheim West
Laupheim
Stadt
Schemmerberg
Riedlingen
Rechtenstein
Biberach (Riß) Süd
Tannheim
Marstetten-
Aitrach
Wolfegg
Wangen (Allg)
Ravensburg
Meckenbeuren
Aichstetten
Bad Waldsee
Bad Wurzach
Warthausen
Finninger-
straße
Gerlenhofen
V
Wullens
B
Illertis
Biberach (Riß)
Bad Schussenried
Bad Saulgau
Altshausen
Ostrach
Burgweiler
Pfullendorf
Alttann
Oberzell
Weißenau
Weingarten/Berg
Niederbiegen
Mochenwangen
Bisingen
Engstlatt
Stetten (b Haigerloch)
Haigerloch
Bad Imnau
Mühringen
Balingen (Württ)
Frommern
Geisingen
Geisingen-Hausen
Geisingen-Kirchen
Geisingen-
Leipferdingen
Geisingen-
Aulfingen
Blumberg-
Riedöschingen
Blumberg-
Zollhaus
Tuttlingen
Schulen
Immendingen Mitte
Möhringen Rathaus
Möhringen Bf
D-Mitte/Siedlung
D-Aufen
D-Grüningen
Brigachtal-Klengen
Brigachtal-Kirchdorf
Marbach West (VS)
Marbach
Ost (VS)
Trossingen
Stadt
Schwenningen (N)
VS Hammerstatt
VS Eisstadion
Zollhaus (VS)
Löffingen
Hüfingen Mitte
D-Allmendshofen
Singen
Landesgartenschau
Weizen
Stühlingen
Eggingen
Engen
Deißlingen
Mitte
Aldingen (b Spaichingen)
Spaichingen Mitte
Endingen (Württ)
(Neckar)
Neufra (Hohenz)
Gauselfingen
Burladingen
Burladingen West
Hausen-Starzeln
Killer
Jungingen
Bad Urach Erms
R
Det
Gammertingen-Europastraße
Schlatt
Schömberg Stausee
Schömberg
(b Balingen)
Rottweil-Göllsdorf
Rottweil-Saline
Rottweil-Neufra
Dotternhausen-Dormettingen
Erzingen (Württ)
Spaichingen
Balgheim
Rietheim (Württ)
Weilheim (Württ)
Wurmlingen Nord
Wurmlingen
Mitte
Mühl-
hausen
Markdorf (Baden)
FN-Kluftern
Uhldingen-
Mühlhofen
FN Landratsamt
FN-Manzell
FN-Fischbach
FN Hafen
Kressbronn
Nonnenhorn
Wasserburg (Bodensee)
Enzisweiler
Langen-
argen
Eriskirch
FN Ost
Friedrichshafen
Flughafen
Kehlen
Löwental
Ko.-Petershausen
Ko.-Wollmatingen
Ko.-Fürstenberg
Reichenau (Baden)
Kreuzlingen
Hegne
Allensbach
Markelfingen
Welsch.-Neu-
hausen
Bräunlingen Bf
Bräunlingen Industriegebiet
Hinterzarten
Denzlingen
Köndringen
Teningen-Mundingen
Eichstetten a.K.
Bötzingen
Königschaffhausen
Kemp
Hbf
Weinfelden
Bülach
Baden
Zurzach
Riehen
Riehen Niederholz
Bad Ur
Senden
Figure 1.4: Excerpt from the ocial 2020 regional rail network map of southwestern
Germany. Although schematic, the octilinear connections between stations roughly fol-
low their original course (even between stations) and respect geographical boundaries
on the underlying schematic map (national and regional borders, rivers, lakes).
and also give a weighted variant, then called MLNCM-W. To improve both the esthetic
quality and the readability of the maps, we also introduce the concept of penalizing line
separations. We prove the NP-hardness of these variants and give two integer linear
programs (ILPs) to solve them to optimality, as well as several heuristic approaches.
We conduct a large experimental evaluation on real-world datasets. The results have
been previously published at the 26th ACM SIGSPATIAL International Conference on
Advances in Geographic Information Systems (GIS 2018) [19]. Out of 38 accepted full
papers, the work was one of six to be considered for the best paper award.
Also in the context of MLNCM, we describe and prove the correctness of several sim-
plication rules that may be applied to the line graph prior to optimization. These rules
are able to completely solve MLNCM and its variants for certain types of input graphs,
and greatly reduce the search space size of the optimization problem. The simplica-
tion rules and their eect on both the heuristic and exact solution of MLNCM have been
previously published in an extended article as part of the ACM Transactions on Spatial
Algorithms and Systems, Vol. 5, 2019 [20]. Using these speedup techniques, we were able
to nd optimal line orderings in the weighted scenario with considering line separations
in under 1 second even for large real-world networks.
We give a novel method for generating octilinear transit maps by nding their opti-
mal image in a special octilinear base grid graph. We describe an ILP for this optimization
problem and also give a heuristic method based on the repeated calculation of shortest
6 Chapter 1. Introduction
paths in this grid graph. To further speed up computation times and to render maps fol-
lowing other layout criteria (e.g. orthoradial maps), we study and evaluate our method
on several alternative base grids: sparse grids, which retain octilinearity but greatly re-
duce the number of grid nodes, and non-octilinear grids (pseudo-orthoradial grids and
hexalinear grids). For small to intermediate networks, our heuristic approach typically
nds an octilinear drawing in under 1 second. We also demonstrate that our method
is able to produce high-quality orthoradial and hexalinear maps. The approximation
method, the corresponding ILP and their evaluation have been previously presented at
the 2020 Eurographics Conference on Visualization (EuroVis 2020) and published as part
of the Computer Graphics Forum, Vol. 39, 2020 [21]. An extension of the method, in-
cluding the formulation of sparse base grids and base grids for alternative layouts, has
been presented at the 17th International Symposium on Spatial and Temporal Databases
(SSTD 2021) and published as part of the conference proceedings [22]. Out of 15 full
papers, the work was one of four to be considered for the best paper award.
Finally, all our methods were implemented in free open-source tools. This includes
our tool pfaedle, a tool for map-matching schedule data given in the GTFS format, and
LOOM (Line-Ordering Optimized Maps), an easily extendable software suite for render-
ing and schematizing public transit maps.
1.3 Preliminaries
This section briey outlines a core set of concepts used throughout this work.
1.3.1 Schedule Data
Schedule data can be understood as a set of vehicle trips. Each trip describes the route of a
single vehicle through the transportation network: which stations are served, and when.
Each trip additionally has a line associated with it, which essentially groups multiple
trips into a single service (for example, bus line 12 from Downtown to Main Station).
We call this set the Bag of Trips.
Denition 1.1 (Bag of Trips). We call a tuple (S, L, Λ, T) a Bag of Trips (BoT). S is a set
of stations s = (n, π ), where n is the station name and π ∈ R
2
is a station coordinate. L
is a set of lines, Λ is a set of vehicle types (e.g. bus, rail, or streetcar). T is a set of vehicle
trips T = (Σ, τ
↓
, τ
↑
, l, λ), where Σ = s
1
, . . . , s
n
, with s
i
∈ S, is the sequence of stops served
by this vehicle, τ
↓
= t
↓
1
, . . . t
↓
n
, with t
↓
i
∈ N
+
0
, is the sequence of arrival timestamps, and
τ
↑
= t
↑
1
, . . . , t
↑
n
, with t
↑
i
∈ N
+
0
, is the sequence of departure timestamps (a trip T ∈ T
arrives at s
i
∈ S at t
↓
i
and leaves at t
↑
i
). l ∈ L is the line of the trip, λ ∈ Λ is its vehicle
type. Note that for s
0
and s
n
, t
↓
= t
↑
.
1.3 Preliminaries 7
A common le format for schedule data is the General Transit Feed Specication
(GTFS) [76]. GTFS consists of several comma-separated text les which hold the set of
trips (trips.txt for meta information like vehicle type and line, stop_times.txt for
the stop sequences), the set of stations (stops.txt) and the set of lines (routes.txt).
Other schedule data formats exist. They are often proprietary, like the HRDF for-
mat [77] (used internally at Deutsche Bahn or the Swiss Federal Railways, among others).
Others are based on national standards, like VDV 452 in Germany [139], or CIF [135]
in the United Kingdom, or following international standards, like the NeTEx format in
Europe [138]. They can usually be translated to GTFS with some eort.
1.3.2 Graphs
Denition 1.2 (Graph). A graph is a tuple G = (V , E), where V is the set of nodes, and
E is the set of edges connecting them.
If edges can only be traversed in one direction, G is called a directed graph (or di-
graph). Otherwise, it is called an undirected graph or simply a graph. For an edge e ∈ E
between two nodes u,v ∈ V , we write e = (u, v) if G is directed and e = {u, v} if G is
undirected. For directed edges, we denote by tail (u, v) = u the tail of edge (u,v) and by
head (u, v) = v its head. An undirected graph G = (V , E) can be modeled with a directed
graph G
0
by adding edges (u, v) and (v, u) to G
0
for each edge {u,v} ∈ E.
Denition 1.3 (Node Degree and Adjacency). For undirected graphs, we say nodes u
and v are adjacent if {u, v} ∈ E and write u ∈ adj(v) and v ∈ adj(u). We denote by the
degree deg(u) the number of nodes adjacent to u. For directed graphs, we say node v is
adjacent to u if (u, v) ∈ E (but not if only (v, u) ∈ E) and write v ∈ adj(u) = adj
+
(u). We
denote by the out-degree deg
+
(u) the number of nodes adjacent to u and dene deg(u) =
deg
+
(u). Additionally, we dene the in-degree deg
−
(u) as the number of nodes u is
adjacent to and denote these nodes by adj
−
(u).
Denition 1.4 (Node and Edge Labeling). Given a graph G = (V , E), a node labeling is a
function V 7→ L which assigns each node v a label l ∈ L. Analogously, an edge labeling
is a function E 7→ L
0
which assigns each node v a label l
0
∈ L
0
.
Transportation Networks
We call a directed graph N = (V
N
, E
N
), embedded in R
2
, and modeling the physical
network for a distinct vehicle type (for example, the road network for buses, the rail
network for trains, the light rail network for trams, or ferry routes) a transportation
network for that vehicle type. In particular, a label function Π : V
N
7→ R
2
embeds each
node on a projection of the globe (we ignore elevation), and a label function P : E
N
7→
(p
1
, . . . , p
n
)embeds each edge using a polyline p
1
, . . . , p
n
. In this work, these coordinates
8 Chapter 1. Introduction
𝑒
𝑓
𝑢
ℎ
𝑔
Figure 1.5: Excerpt from a linegraph G =
(V , E, L, L) for the Chicago light rail network
(corresponding to the Chicago loop shown in
Figure 1.2). A labeling function L assigns each
edge a set of lines, with L(e) ⊆ L, depicted
here as colored circles (here, L = { }, and e.g.
L
(
e
)
= { }). In this example, edges are addi-
tionally labeled by a polyline describing the edge
course, see for example edges д and h. Nodes are
usually - but not necessarily - stations. For ex-
ample, u does not correspond to a station (com-
pare Figure 1.2).
are assumed to be in the Web Mercator projection (EPSG:3857), although we often give
them as WGS 84 latitude/longitude pairs. For example, in a network for buses, the edges
E
N
correspond to road segments, and the nodes V
N
correspond to intersections and/or
road anchor points. The graph is directed to model unidirectional segments (e.g. one-
way streets).
A node subset S ⊆ V
N
is maintained which describes the stations of the network. A
node labeling function N (v) assigns each node v ∈ S a set of station labels (e.g. ‘London
St Pancras‘ and ‘St Pancras Station‘).
Line Graphs
The fundamental structure used in this work is a line graph, which can be understood as
the skeleton of a public transit map. Note that “line” here refers to a public transit line.
In graph theory, a graph L(G) that is constructed by converting each edge in G into a
node in L(G) is also often called a line graph [80]. In the context of this work, we will
call such a graph L(G) the edge-to-node dual of G. It will be introduced in more detail in
Section 2.6.4.
In its most basic form, a line graph G = (V , E, L, L) is an undirected labeled graph
with nodes V and edges E (weakly) embedded in R
2
. L is an edge labeling function
which assigns an edge e ∈ E a set of lines L(e) ⊆ L, where L is the set of all lines. A
set S ⊆ V describes station nodes, and the V \ S are called non-station nodes. Station
nodes correspond to stations in the nal map, whereas non-station nodes are typically
places where lines branch. Just as for transportation networks, we often assume a label
function n which assigns each node s ∈ S a station label.
We also often use a function R which assigns pairs (e, f ) of adjacent edges and some
1.3 Preliminaries 9
line l ∈ L(e) ∩ L(f ) a line turn restriction. That is, we mark by R(l) = (e, f ) that the line
l does not continue from e to f (in the context of this work, we ignore the case where
a turn restriction involves more than 2 edges). This will be described in more detail in
Chapter 3.
1.3.3 Shortest Paths
Denition 1.5 (Edge Weights). We call a graph G weighted if G has an edge labeling
w : E 7→ R
+
0
(we are not interested in negative weights).
Denition 1.6 (Path). A sequence of edges e
1
, e
2
, . . . , e
n
is called a path if there is a
corresponding sequence of nodes v
1
,v
2
, . . . , v
n+1
such that e
i
= (v
i
,v
i+1
). A path p is
called simple if the sequence of nodes is distinct (this implies that a circular path is not
simple). A single node v is a simple path. We denote by |p| the number of edges in the
path.
Denition 1.7 (Path Cost). If p is a path in a weighted graph, it has a cost c(p) =
n
i=1
w(e
i
). The cost of a single-node path is 0.
Denition 1.8 (Shortest Path). A path p between nodes s and t in a graph G is a shortest
path if no other path p
0
exists with c(p
0
) < c(p).
Lemma 1.1. A shortest path in a graph with edge weights > 0 is simple.
Proof. Assume there is a shortest path p from s to t which visits a node v twice. If we
remove the cycle fromv tov, the resulting path p
0
still goes from s to t. Because w(e) > 0,
c(p
0
) < c(p), a contradiction.
Dijkstra’s Algorithm Given a directed graph G = (V , E) with non-negative edge
weights and some start node s ∈ V , the shortest path from s to all nodes V \ {s} can
be computed using Dijkstra’s algorithm [50]. The algorithm can be considered an itera-
tive implementation of a dynamic programming approach [131] based on the following
intuition: the shortest path from s to some node t is the shortest path from s to a node
v ∈ adj
−
(t) (with cost c
v
), plus the edge (v, t). Under all nodes adj
−
(t) this is precisely
the node v for which c
v
+ w((v, t)) is smallest. This immediately gives rise to a recur-
sive computation scheme in which the subproblems show great overlap (we only have
to compute the shortest path to some particular node w once, but it may appear very
often during the recursive computation). In the classic iterative implementation of Di-
jkstra’s Algorithm, each node v is initially assigned a distance label c
v
= ∞, except the
start node s, which is labeled with c
s
= 0. A priority queue Q is maintained that holds
all nodes with their distance labels so far, initially only containing s with c
s
= 0. The
algorithm then repeatedly takes the node v with lowest c
v
from Q and updates the labels
10 Chapter 1. Introduction
of each adjacent node u with c
0
u
= c
v
+ w((v, u)) if c
0
u
< c
u
. v is said to be expanded and
u is said to be relaxed. An important property is that each time a node v is extracted
from Q, the cost label c
v
is the cost of the shortest path from s to v. The extracted node
is hence called settled. A pointer to the predecessor node from which the last relaxation
came from may be maintained, allowing the later reconstruction of the shortest path.
A typical variant of Dijkstra’s algorithm is to search for the shortest path from a sin-
gle source node s to a single target node t, and in this case, the search can be aborted if t is
settled (or if the priority queue runs empty, in which case there is no path from s to t). If a
Fibonacci heap is used to implement the priority queue [64], Dijkstra’s algorithm runs in
time O(|E| + |V | log |V |). No asymptotically faster algorithm for computing the shortest
path on general graphs is known. There are, however, many established speedup meth-
ods used in practice. They range from ecient implementation techniques of Dijkstra’s
algorithm, over heuristics which prefer certain nodes during the expansion process and
thus steer the shortest path search towards the goal [141], to elaborate preprocessing
schemes which later speed up the shortest path calculation, sometimes dramatically.
A typical example for an implementation-based speedup technique is to use a special
monotone priority queue (for example a radix heap [3]), or to perform the search from
both s and t simultaneously until the search barriers meet (Bidirectional Dijkstra). A
typical example for a heuristic approach is the A
∗
algorithm [81], which can be consid-
ered an extension of Dijkstra’s algorithm. Some well-known examples for preprocessing
schemes are Landmark heuristics (in which a special A
∗
heuristic is precomputed [74])
and Contraction Hierarchies [70]. We only describe A
∗
in detail here, as it will be used
several times in the following chapters. For a thorough survey on various speedup tech-
niques for nding shortest paths in transportation networks, we refer to [23].
A
∗
Algorithm Instead of adding a node v with its label c
v
as described above as a
key to the priority queue Q, we instead use c
v
+ h(v), where h(v) is a heuristic function
which estimates the cost from v to the target node, thus preferring nodes which are
heuristically closer to the target during the expansion process. Obviously, if h(v) = 0
for each v ∈ V , this is exactly Dijkstra’s algorithm. There are two important criteria
the heuristic function can meet and which are often confusingly mixed: (1) admissibility
and (2) consistency. A heuristic function h is said to be admissible if ∀v ∈ V : h(v) ≤
c(v, t), where c(v, t) is the shortest path cost from v to t. In other words, h must never
overestimate the actual cost to the target. A heuristic function is said to be consistent if
for every adjacent pair of nodes v, u it holds that h(v) ≤ w((v,u)) + h(u) (an equivalent
requirement is h(v) ≤ c(v, u) + h(u), where c(v, u) is again the shortest path cost from
v to u). In other words, h must never overestimate the cost to the target in such a way
that if we take the edge cost to the adjacent neighbor u into account, u would provide
a better overall estimate. A consistent heuristic is always admissible, but an admissible
heuristic is not necessarily consistent.
1.3 Preliminaries 11
If h is consistent, then at the point of expanding v, the cost label c
v
is the cost of the
shortest path from s to v, just like with the classic Dijkstra’s algorithm (see below for
a proof). If h is not consistent, but only admissible, this is not guaranteed anymore. A
node might thus require multiple expansions.
If the implementation allows such a repeated expansion of nodes, then A
∗
returns the
shortest path from s to t with a heuristic that is admissible, but not consistent. Impor-
tantly, the target will never require multiple expansion with an admissible, but inconsis-
tent heuristic, and the algorithm can hence be terminated as soon as t is expanded:
Lemma 1.2. Given an inconsistent, but admissible heuristic h. If we expand the target node
t, c
t
is guaranteed to be the cost of the shortest path from s to t, and we have thus found
this shortest path.
Proof. As we have extracted t with label c
t
from Q, there is no node v with c
v
+ h(v) <
c
t
+ h(t) = c
t
. Let c be the cost of the actual shortest path from s to t, and assume c
t
was
not optimal, that is c
t
> c. Then there is some node v ∈ V (this might be s itself!) with
c
v
+ c(v, t) < c
t
, where c(v, t) is the shortest path cost from v to t. But as h is admissible,
h(v) ≤ c(v, t), and thus c
v
+ h(v) < c
t
+ h(t), a contradiction.
Crucially, with an inconsistent but admissible heuristic, this proof only works for the
target node t. For a consistent heuristic, it works for any node u: if u is expanded, the
existence of some node v with c
v
+ c(v, u) < c
u
together with the consistency require-
ment h(v) ≤ c(v, u) + h(u) would imply c
v
+ h(v) < c
u
+ h(u), which again leads to a
contradiction. c
u
was thus the cost of the shortest path to u.
To summarize: With a consistent heuristic, A
∗
produces the shortest path from s
to t and only expands each node at most once. With an inconsistent, but admissible
heuristic, A
∗
might require nodes to be expanded multiple times, but if t is reached, we
are guaranteed to have found the shortest path.
One-To-Set and Set-to-Set Shortest Paths A simple variation of Dijkstra’s algo-
rithm may be used to compute the shortest path from a single source s into a set of
target nodes T (that is the shortest path from s to any of the nodes in T ): we simply
abort the algorithm as soon as a node t ∈ T is settled, or if the priority queue runs
empty (in which case, again, there is no path from s to T ).
Another variant for which the correctness is not immediately obvious are set-to-set
shortest paths. Here we are interested in nding the shortest path from any node in a
source node set S to any node in a target node set T .
Lemma 1.3. If the priority queue Q in Dijkstra’s algorithm is initialized with a set of nodes
S, each with cost label 0, the algorithm will produce the shortest paths from any node in S.
12 Chapter 1. Introduction
Proof. We add a virtual node v
∗
and connect v
∗
to each node s ∈ S with an edge of
weight zero. The shortest path p from v
∗
to any other node will contain a node s ∈ S,
and the c(p) will be the cost of the shortest path starting from this node s (as the initial
edge from v
∗
has weight 0). This will be the node from S from which the shortest path
starts: if this would not be the case, there would be a path p
0
from v
∗
over a dierent
node s
0
∈ S with c(p
0
) < c(p), but p was already the shortest path. Adding all nodes S to
Q with cost label 0 is exactly the state of Q after the rst relaxation of all adjacent edges
to v
∗
, and the initialization thus implicitly constructs v
∗
and its edges to S.
One-to-Many Shortest Paths The one-to-many shortest path problem searches for
the shortest paths (or decides that a path does not exist) from a single source node s to all
nodes in a target set T . It is evident from the algorithm description above that a standard
Dijkstra implementation already computes a superset of the desired shortest paths and
we can simply stop the algorithm as soon as all nodes t ∈ T have been settled.
Set-to-Many Shortest Paths Similar to the set-to-set shortest path problem is the set-
to-many shortest path problem, which searches for the shortest path from any source
node s ∈ S to all nodes in a target set T . Obviously, the same technique as for the set-to-
set case can be used, but instead of aborting the search as soon as a node in S is settled,
we let it run until all nodes in S are settled, or until the priority queue runs empty.
Many-to-Many Shortest Paths Finally, the many-to-many shortest path problem
searches for all pairwise shortest paths from a set of source nodes S to a set of target
nodes T . A naive implementation would compute these paths by |S | · |T | Dijkstra runs.
A more ecient way is to compute |S | one-to-many shortest paths from a node s
i
to |T |.
1.3.4 Optimization
Finding the shortest path through a graph is an optimization problem: we want to nd
the path that optimizes the cost function. In general, an optimization problem searches
for the best solution from some solution space Ω. This solution space is often called the
set of feasible solutions, or feasible set. The set of feasible solutions is dened by a set of
constraints. A feasible solution (consisting of possibly multiple values x
1
, . . . , x
n
) x ∈ Ω
is considered better than a feasible solution y ∈ Ω if the objective function f : Ω 7→ R
gives f (x) < f (y). The objective function f is also often called loss function, cost function,
or target function. The optimization problem may also be called a minimization problem,
as the goal is to minimize f . Each minimization problem can be transformed into a
maximization problem (and vice versa) by using an objective function f
0
= −f .
1.3 Preliminaries 13
Local Search
Local search is a widely used class of heuristic optimization algorithms. For feasible
solutions x ∈ Ω, a neighborhood N (x) ⊂ Ω is dened which consists of all feasible
solutions x
0
which can be obtained by making a small change to x. The exact denition
of the neighborhood is highly domain specic. Given the current solution s ∈ Ω, all
s
0
∈ N (s) are then evaluated in terms of their objective function, and one of them is
chosen as the new solution candidate. How this new solution candidate is selected is
the distinguishing feature of these algorithms. For example, if from all N (s) the s
0
which
gives the greatest improvement is chosen, this is called Steep est-Ascent Hill Climbing.
Simple Hill Climbing always selects the rst better neighbor found, and Stochastic Hill
Climbing examines random neighbors until a better one is found. Such greedy local
search variants typically get stuck in a local optimum. Some variants like Simulated
Annealing [120] hence introduce chaos into the system to push the search out of local
optima.
Linear Programming
If both the objective function f and the constraints of an optimization problem are linear,
the optimization problem is called linear programming (LP). The objective function can
then be expressed as f (x) = c
T
x, where c is a vector of coecients. The constraints
can be expressed by a vector b of coecients (the right-hand side of the constraints),
and a matrix A of coecients (the left-hand side of the constraints, each row holds one
coecient for each solution variable).
Denition 1.9 (Canonical Linear Program). Given c, b, and A:
max
c
T
x | Ax ≤ b ∧ x ≥ 0
. (1.1)
While the canonical form seems to restrict linear programs to maximization prob-
lems with non-negative variables and only less-than-or-equal constraints, minimization
problems, negative variables, greater-than-or-equal constraints, or equality constraints
can all be expressed in it.
As mentioned above, any minimization problem can be turned into a maximization
problem by multiplying the coecients of the objective function c by −1. Similarly, to
get greater-than-or-equal constraints, the constraint can be multiplied by −1. Equality
constraints of the form a
i
x = b
i
can be modeled by two constraints a
i
x ≤ b
i
and a
i
x ≥ b
i
.
Finally, a free (positive or negative) variable x
i
can be modeled by two non-negative
variables x
0
i
and x
00
i
and substituting x
i
by x
0
i
−x
00
i
.
In its standard form, the unknown variables of a linear program are real numbers,
that is x
i
∈ R. There are several forms with additional constraints on the variables:
14 Chapter 1. Introduction
Denition 1.10 (Integer Linear Programming). If the unknown variables of an LP are
integers, the problem is called integer linear programming, and a problem instance is
called an integer linear program (ILP). Solving an ILP without the integrality constraints
is called LP-relaxation and may be used to compute a lower or upper bound for the target
function.
Denition 1.11 (Binary Programming). If the unknown variables are additionally re-
quired to be binary, the problem is called binary integer programming (BIP) (it is easy to
see that a BIP can be modeled via an ILP by adding additional constraints to all integers
which restrict their values to 0 and 1).
Denition 1.12 (Mixed Integer Programming). If the unknown variables can be either
integers or real numbers, the problem is called mixed integer programming (MIP).
While polynomial algorithms for solving linear programs exist (the rst was de-
scribed by Khachiyan in [93], the well-known Simplex algorithm by Dantzig [43] has
exponential worst-case running time [95]), the optimization of integer linear programs,
binary programs and mixed integer programs are all NP-hard [67]. In practice, ILPs are
usually optimized by special solvers. Their performance has typically been optimized
for decades, and they employ a wide set of sophisticated heuristics. Prominent exam-
ples (in ascending order of general performance) are the GNU Linear Programming Kit
(GLPK)
2
, COIN-OR CoinMP
3
, and gurobi
4
. Both GLPK and CoinMP are free to use.
2
https://www.gnu.org/software/glpk/
3
https://www.coin-or.org/projects/
4
https://www.gurobi.com/
Chapter 2
Map-Matching Schedule Data
Map-matching is the process of matching a sequence of sample points (e.g. GPS traces)
to a geographical model of the real world (e.g. a road network). Map-matching schedule
data means nding the most likely path a public transportation vehicle takes on a sched-
uled trip through its transportation network, given only the vehicle stops (consisting of
a geographical coordinate and a station label) as sample points (Figure 2.1). These paths
are often called shapes. Formally, this chapter considers the following problem:
Problem 1 (Map-Matching Schedule Data). Given a Bag of Trips (S, L, Λ, T) and trans-
portation networks N
λ
for each vehicle type λ ∈ Λ, nd the most likely path p (its shape)
through the respective N
V
for each trip T ∈ T. For each T , p must consist of |Σ| − 1
consecutive parts p
1
, p
2
, ..., p
|Σ|−1
, where p
i
is the path from stop s
i
to stop s
i+1
.
Given the high relevance of this problem for static or dynamic visualization of public
transit systems or for real-time map-matching of mobile devices to moving trains, map-
matching of schedule data has received only little research attention so far. In particular,
if we consider the high spatial sparseness of the sample points (the station coordinates),
it is not clear at all whether existing approaches for map-matching (typically dense) GPS
traces to road networks are applicable.
After a motivation and some preliminaries, we provide an overview over related
work. Section 2.4 then describes several baseline approaches to this problem, followed
by a description of a hidden Markov model (HMM) to nd the most likely path (Sec-
tion 2.5). This model only uses the spatial information contained in the schedule data.
We show how the edge-to-node dual graph of the transportation network can be used
to naturally include turn restrictions both for individual transitions, and between transi-
tions in Section 2.6. Section 2.7 describes how the map-matching results can be improved
by incorporating additional metadata from the transportation network. Here, we also
describe methods to decide whether two station identiers describe the same real-world
station. Section 2.9 then outlines several speedup techniques. We close with an experi-
mental comparison of our approaches in Section 2.10.
15
16 Chapter 2. Map-Matching Schedule Data
InterCity 729
Station Arr. Dep. Lat. Lon.
1 Genève Aero. — 16:32 46.231 6.111
2 Genève 16:42 16:44 46.211 6.142
3 Lausanne 17:20 17:20 46.517 6.629
4 Fribourg 18:04 18:04 46.803 7.151
5 Bern 18:32 18:33 46.949 7.438
6 Zürich 19:33 19:36 47.378 8.539
7 Zürich Flugh. 19:44 19:45 47.451 8.562
8 Winterthur 19:59 20:00 47.501 8.723
9 St. Gallen 20:35 — 47.423 9.369
Figure 2.1: Left: Schedule data for a single trip through Switzerland. Top right: Shape
obtained by the station coordinates. Bottom right: Shape map-matched by our approach.
2.1 Motivation
Our main motivation for Problem 1 is the generation of geographically accurate transit
maps as described in Chapter 1. However, there are many other applications, which we
briey discuss in this section. As an additional motivation, we also conducted a survey
of the general availability and quality of shapes in publicly available schedule data. The
methodology and results are presented in Section 2.1.2.
2.1.1 Applications
Apart from the need for high-quality shapes in geographically accurate transit maps,
Problem 1 is an important preprocessing step for many other areas which use public
transit data. For example, exact shape data is also valuable for schematic maps, as it
allows the identication of segments shared by multiple lines.
Route-planning systems for public transportation typically also visualize the calcu-
lated route on a map. For this visualization to be helpful, the shapes of the individual ve-
hicle trips which make up the route must be known. Figure 2.2 gives three examples from
the route planners of Google Maps, Deutsche Bahn, and Bing Maps where shape data
was missing. The routes (from Freiburg to Berlin) are displayed with straight lines be-
tween stations and oer only little additional value. Some public transit agencies already
oer interactive maps which show the live position of each vehicle in their network, and
there are several such maps covering more than one agency or entire countries (e.g. the
2.1 Motivation 17
Figure 2.2: Visualization of public transit routes in Google Maps (left), the ocial
Deutsche Bahn route planning service (middle) and Bing Maps (right).
MTA Live Subway for New York
1
, or our own work TRAVIC
2
). As real-time public tran-
sit data often only provides delay times, the vehicle shapes must be known to determine
the current position. Even if exact vehicle positions are delivered, the knowledge of the
vehicle shapes is advantageous. Consider, for example, a stream of live vehicle posi-
tions delivered by a GPS device on a moving bus. The positions may only be updated
every few seconds or fail completely if the vehicle enters a tunnel. To ensure a smooth
visualization in situations like this, the scheduled vehicle path is valuable.
Another interesting problem for which exact shape information is important is that
of matching mobile devices to moving trains. A typical scenario is a smartphone app
showing information regarding the train the owner is currently sitting it, but it may also
be used to crowdsource delay data.
Lastly, map-matching of schedule data may be used to improve the quality of the
network data itself. We can very easily use the map-matched paths to augment the
original edge in the transportation network with information about the vehicle trip. For
example, OSM oers the concept of public transit line relations to map line courses
3
.
These may be lled automatically using this approach.
2.1.2 A Survey on Shape Availability and Quality
To further motivate our work, we conducted an extensive survey of the shape quality of
publicly available schedule data. With over 900 published feeds in 2020 (see Table 2.1),
https://transitfeeds.com is the largest collection of public transportation schedules
we know of. The schedules are published as GTFS feeds. All feeds are regularly updated
and archived. The archive can be queried using a simple API.
1
https://map.mta.info
2
https://travic.app
3
https://wiki.openstreetmap.org/wiki/Public_transport
18 Chapter 2. Map-Matching Schedule Data
𝑠
1
𝑠
2
𝑠
3
𝑠
4
𝑠
5
𝑠
6
𝑠
7
𝑦
𝑥
𝑇
𝑠
1
𝑠
2
𝑠
3
𝑠
4
𝑠
5
𝑠
6
𝑠
7
𝑦
𝑥
𝑑
1
𝑑
2
𝑇
250𝑚
Figure 2.3: Left: A valid shape (red) for a trip T visiting stations s1, s2, s3, s4, s5, s6, s7, in
that order. The dashed black line is the as-the-crow-ies shape we consider degenerate.
Right: an erroneous shape (red) for the same trip. The distance d
1
between station s
1
and the shape is greater than the 250 meters threshold. The same holds for distance d
2
.
For each year beginning with 2015, we downloaded all GTFS feeds that were active
on March 1st of that year. We chose March 1st because many agencies publish new
major versions of their timetables at the beginning of a new year. We assumed that this
transition phase is over in spring. For each vehicle trip in this data, we evaluated the
quality of the associated shape. We assigned the following categories to each trip:
NO-SHP The vehicle has no associated shape information.
ERR One or more stops of the vehicle trip had geographical coordinates with a dis-
tance greater than 250 meters to the associated shape. This error may occur in the
following situations: (1) The assigned shape is the shape for another trip. (2) The
trip has been updated in the past to serve dierent stops, but the shape was not
updated accordingly. (3) The trip was assigned a “default” shape for a line, but has
a slightly dierent stop sequence than the majority of trips traveling on the line
(e.g. an outlier trip during rush hour, or because the trip is traveling back to some
depot).
DEG The associated shape was degenerated. We say a shape is degenerated if it has ex-
actly one anchor point for each served station. In other words, the shape is exactly
the default as-the-crow-ies representation. In rare cases, the correct geographical
path between each subsequent stop pair in a trip’s stop sequence may be a straight
line, which would then be falsely classied as degenerate.
OK None of the above apply, the shape is considered to be of good quality.
The results can be seen in Table 2.1. Both the number of feeds and the total num-
ber of trips contained in all of them have more than tripled since 2015. However, the
percentage of trips that were provided with shape information decreased over the years
investigated, from 60.4% in 2015 to 58.8 in 2020, further establishing the need for an
2.2 Preliminaries 19
Table 2.1: Dimensions and shape quality of GTFS feeds published on https://
transitfeeds.com between 2015 and 2020. Under “Feeds”, we give the total number of
feeds contained (TOT) and the number of feeds with shapes (HAS-SHPS). Under “Trips”,
we give the number of trips contained in all feeds (TOT), the number of trips with no
shape (NO-SHP), the number of trips with erroneous shapes (ERR), the number of trips
with degenerated shapes (DEG) and the number of trips with good shapes (OK).
Feeds Trips
TOT HAS-SHPS TOT NO-SHP ERR DEG OK
2015 292 258 5.1M 1.9M (39.1%) 9.5k (0.2%) 22.9k (0.5%) 3.1M (60.4%)
2016 440 399 6.4M 2.2M (34.7%) 20.7k (0.3%) 26.9k (0.4%) 4.1M (64.6%)
2017 642 573 11.2M 4.1M (36.5%) 50.0k (0.5%) 49.1k (0.4%) 7.1M (62.6%)
2018 667 605 11.7M 4.0M (34.3%) 53.5k (0.5%) 45.3k (0.4%) 7.6M (64.9%)
2019 708 653 11.9M 4.1M (34.1%) 33.5k (0.3%) 94.3k (0.8%) 7.7M (64.8%)
2020 906 805 17.8M 7.3M (40.4%) 88.4k (0.5%) 61.3k (0.3%) 10.5M (58.8%)
automated solution to generate these shapes. The setup for this survey can be found
online
4
.
We note that a particularly big schedule dataset is missing from transitfeeds.com:
the entire schedule data of Germany. This dataset has recently been published in the
NeTEx format and converted to GTFS. It contains over 2.3 million trips (more than 10 %
of the number of trips contained in transitfeeds.com) and lacks any shape data. Based
on repeated contact we had with public transportation authorities in Germany, it is our
belief that such a nationwide schedule dataset with shape information does indeed not
exist. This would make the dataset we generated in our experimental evaluation the rst
of its kind for an entire nation the size of Germany.
2.2 Preliminaries
This section introduces some additional concepts and denitions used in this chapter.
2.2.1 Probability Theory
Denition 2.1 (Events and Sample Spaces). A sample space is a set Ω of outcomes
s
1
, s
2
, . . . , s
n
of an experiment. An event E ⊆ Ω is a subset of these outcomes. For
example, the sample space of rolling a die is {1, 2, 3, 4, 5, 6}, and A = {2, 4, 6} is the event
of throwing an even number.
4
https://github.com/ad-freiburg/gtfs-shape-quality-survey
20 Chapter 2. Map-Matching Schedule Data
Denition 2.2 (Probability Distribution). A probability distribution describes the prob-
abilities of events in a sample space. Pr(X = x)is the probability that the random variable
X takes on value x. If X is discrete and can only take values x
1
, . . . , x
n
, the probability
distribution is given as a non-negative probability mass function p(x) = Pr(X = x) with
Σ
n
i=1
p(x
i
) = 1. If X is continuous, its distribution is given as a probability density func-
tion.
Denition 2.3 (Probability Density Function). The density of a continuous random vari-
able X is a non-negative function f (x) for which holds that (1)
∞
−∞
f
X
(x)dx = 1 and (2)
b
a
f
X
(x)dx = Pr(a ≤ X < b).
Denition 2.4 (Cumulative Distribution Function). The cumulative distribution func-
tion of a continuous random variable X is the function F
X
(x) = Pr(X ≤ x) =
x
−∞
f
X
(y)dy
that gives the probability that X is smaller than or equal to x. In particular, 1 −F
X
(x) =
Pr(X > x). As Pr(X = x) = 0 for any continuous random variable X and any x,
Pr(X ≤ x) = Pr(X < x) and Pr(X ≥ x) = Pr(X > x).
Denition 2.5 (Conditional Probabilities). Given two events A and B, the conditional
probability Pr(A | B) is the probability that A occurs given that B already occurred.
2.2.2 Hidden Markov Model
As mentioned above, our approach is based on a hidden Markov model (HMM). It may
be dened as follows:
Denition 2.6 (Hidden Markov Model). A hidden Markov model (HMM) is a tuple
(H, O, T, E, q
0
) where H = {h
1
, . . . , h
n
} is the set of hidden states, O = {o
1
, . . . , o
m
}
is the set of observations, T = {a
ij
}, a
ij
= Pr(h
j
| h
i
) gives the state transition probability
distributions (the probability from going from state h
i
to state h
j
), and E = {b
ij
}, b
ij
=
Pr(o
j
| h
i
) gives the emission probability distributions (the probability that state h
i
emits
observation o
j
). q
0
∈ S is an initial state.
The initial state is sometimes omitted and an initial state probability distribution is
given instead [124]. If an initial state q
0
is dened, the transition probabilities from q
0
to any other state can be understood as this initial state distribution. One may now ask:
given an observation sequence S = s
1
, s
2
, . . . , s
N
, where s
k
∈ O, what is the most likely
sequence of hidden states i
1
, i
2
, . . . , i
N
, i
k
∈ H? Formally, we want to nd
arg max
i
1
,... ,i
N
N
k=1
Pr(i
k
| i
k−1
) · Pr(s
k
| i
k
). (2.1)
An often used O(|O | · |H |
2
) algorithm for this problem is the Viterbi algorithm [140].
2.3 Related Work 21
2.3 Related Work
Our work is closely related to previous work on map-matching GPS traces and schedule
data, routing with turn restrictions, curve similarity measures (used in our experimental
evaluation), string label similarity, and similarity measures for geographic locations.
2.3.1 Map-Matching of GPS Traces
The goal of map-matching GPS traces is to nd a sequence of edges in a transporta-
tion network (usually a road network) that best matches a sequence s
1
, . . . , s
n
of sample
points from a GPS device. A sample s
k
may simply be the location measurement re-
ported by the GPS device, but may also contain measurements regarding the current
vehicle speed, orientation, elevation, or timestamps.
Map-matching of GPS traces is often divided into two categories: In oine map-
matching, the sequence of sample points s
1
, . . . , s
n
is completely known at the time of
map-matching. This is for example the case if measurements were taken during a trip
and map-matched after the trip was completed. In this scenario, a major advantage
is the knowledge of driver intent - it is known from the sample point sequence where
the vehicle was headed, and this information may be included in the map-matching
approach.
In online map-matching, the sequence of sample points s
1
, . . . , s
n
is not completely
available at the time of map-matching, but only as a prex up to a certain s
i
. New mea-
surements are continuously added to the sequence prex. A typical example is a naviga-
tion device in a traveling car, where the current vehicle position and the route traveled so
far are displayed on a map. Besides the lack of information about future sample points,
the major challenge in online map-matching is the computation time.
Because of the high relevance of this problem for navigation systems, a vast amount
of approaches has been described in the literature so far. For a recent survey, see for
example [37]. [145] categorized map-matching approaches into three broad classes: (1)
incremental max-weight, (2) global max-weight, and (3) global geometric. Both incre-
mental and global max-weight methods consist of three steps: (1) for each sample point
s
i
, a set of candidate locations (e.g. road segments or graph nodes) is determined. (2) A
weight is calculated for each of these candidate locations (e.g. the distance to the sample
point, or a weight based on a comparison of the current candidate location to a previ-
ously chosen one). (3) The candidate sequence with the maximum weight is determined.
Incremental max-weight methods greedily optimize the candidate sequence, and global
max-weight methods nd the globally optimal candidate sequence.
If max-weight based methods model the weights as probabilities and aggregate them
by multiplication (essentially calculating the candidate sequence of highest probability)
it is often called a probabilistic model. Typically, a probability distribution is dened
22 Chapter 2. Map-Matching Schedule Data
c
s
1
s
2
b
a
Figure 2.4: A typical problem of HMM based methods which only dene non-zero tran-
sition probabilities to adjacent road segments. The red dots are sample points, the black
lines are road segments, the black nodes are road intersections, and the most likely path
through the network is depicted in blue. With a suciently high sampling rate, the cor-
rect path through the network is found (left). When confronted with very low sampling
rates (right), such methods are typically not able to interpolate between unadjacent road
segments. Road segment b is chosen as the candidate segment for sample point s
2
, al-
though c clearly would’ve been the better choice - but the transition probability from a
to c is zero.
which gives the probability that a location candidate in the road network is correct for
some measurement s
i
. This is usually combined with a probability distribution which
gives the probability that a location candidate is the successor of a previous candidate.
Regarding probabilistic models, approaches based on a hidden Markov model (HMM)
have received considerable research attention in the past. Each candidate location is then
a hidden state h in the HMM according to Denition 2.6, and each sample point s is an
observation. The emission probabilities model how likely it is for a candidate location to
be the correct choice for a sample point. The transition probabilities model how likely
it is for a candidate location h to be the successor of a candidate location h
0
.
In [86] and [121], a map-matching approach based on a HMM was described which
uses road segments as position candidates. Sample points consisted of GPS location
measurements and vehicle orientations. The emission probabilities were modeled on (1)
the angular dierence between the vehicle orientation and the road segment orienta-
tion and (2) on the Euclidean distance between the GPS location and the road segment.
Those distances (essentially the measurement errors) were assumed to follow a zero-
mean Gaussian. Transition probabilities were modeled uniformly, with special penal-
ties for transitions describing a U-turn. Non-zero transition probabilities between road
segment candidates were only given to segments which shared a common node. This
imposes a subtle, but important restriction: namely, that each road segment candidate
must be “selected” by at least one sample point s to appear in the nal map-matched path.
It is not possible to “jump” over multiple road segments. Such a segment interpolation,
however, is a common requirement for sparse sample points.
2.3 Related Work 23
In [109] and independently in [104], this problem was identied and the idea of us-
ing shortest path calculations to both determine the transition probabilities and to in-
terpolate between road segment candidates was developed. In both works, the emission
probabilities were modeled only on the distance between the road segment candidate
and the sample point, again assuming a zero-mean Gaussian distribution of the GPS
noise. The transition probabilities were determined as follows: for each candidate h
j
,
consisting of a road segment r
j
and a position p
j
on this road segment, a shortest path
query was run between p
j
and the position p
j
0
on the next candidate h
j
0
. The transition
probability was then modeled in [109] based on the dierence between the shortest path
distance between p
j
and p
j
0
in meters and the as-the-crow-ies distance between p
j
and
p
j
0
in meters. In [104], the ratio of these distances was chosen. Additionally, [104] based
the transition probabilities on the similarity of the speed pattern between the sample
points and the speed limits of the road segments. This may be error-prone, in particular
because vehicle speed patterns may dier greatly from the speed limits of the used road,
e.g. because of heavy trac. With a sampling rate of 120 seconds, [109] report a ratio
between the length of incorrectly matched segments and the total length of the correct
route of around 0.16.
Eisner et al. [52] used a global max-weight approach which optimizes the length
of the path connecting candidate sets for the sample point (selected using a threshold
distance). To nd the optimal path, a layered graph resembling the state graph found
in HMM based approaches was used. A speedup technique was described which allows
the calculation of all required shortest paths between two candidate sets with a single
run of Dijkstra’s algorithm. In Section 2.9.1, we extend this technique to also work in
the HMM scenario, where emission probabilities must also be considered.
In [132], an approach was introduced which generates k path candidates by rst
calculating a shortest path from s
1
to s
n
and then running k − 1 additional shortest path
searches from s
1
to s
n
, where in each new search, the longest road-segment used in the
previously found shortest path is added to a set of blocked road-segments. At the end,
the path candidate with the smallest summed distance to all sample points s
1
, ..., s
n−1
is chosen. Such an approach is obviously restricted to cases where the map-matched
vehicle was following a shortest path from s
1
to s
n
. This may be true in a goal-directed
private car scenario (or a taxicab scenario, as analyzed by the authors), but not for a
public transit vehicle, where the general priority is to serve as many people as possible
en-route, not to nd the shortest path from the rst to the last station.
Some works also try to improve the map-matching quality under low sampling rates
by leveraging additional meta-information. For example, in [119], a learning-based ap-
proach was used to analyze the number of turns relative to the travel distance tolerated
by a human, which was then used during the map-matching. Zheng et al. [156] ana-
lyze historical trajectory data to derive travel patterns and use this data to suggest likely
paths. Aly and Youssef [8] describe a HMM based approach that incorporates road se-
24 Chapter 2. Map-Matching Schedule Data
mantics detected by smartphone data (e.g. sharp turns, tunnels, bumpy roads).
A recent max-weight approach considers the case of semi-restricted trajectories [24].
These are trajectories for which the assumption that they follow the underlying trans-
portation network is only partially correct, which may for example be the case if pedes-
trians cross large squares, or hike cross-country.
As the name tells, the max-weight methods described so far use weight parameters,
which may require signicant tuning. In contrast, global geometric approaches only use
geometric similarities between the underlying network and the GPS trace to nd the
most likely path. Early approaches used the network path with the minimum Fréchet
distance [63, 5, 30] by extending a method from [6] to graphs. In [30], the weak Fréchet
distance was used instead, producing exactly the same paths in experiments (however,
the weak Fréchet distance uses a non-monotonic parametrization and is hence not able
to consider edge directions). This approach was further improved in [146].
Our method is also based on a hidden Markov model, and is therefore a global max-
weight approach. The main dierences between our work and existing work are: (1)
The notion of what is considered sparse input data. In existing work, sampling rates
of over 30 seconds [109], with an average distance between sample points of around
400 meters (assuming a vehicle travels with an average speed of 50 km/h) were already
considered sparse. In [156], which specically targeted sparsely sampled GPS traced, a
sampling rate of over 2 minutes was considered sparse. In comparison, we have to handle
cases where the average distance between sample points may be up to several hundred
kilometers (for example, for long distance trains). (2) The method of transportation of
the underlying network. Previous work on map-matching was mostly limited to road
networks. In a public transit scenario, we must be able to do map matching also on
rail or ferry networks. Here, physical limitations (for example, regarding acute turn
angles) inherent to both the network and the vehicle type must be considered to achieve
reasonable matching results. (3) The notion of what can be considered as candidates in
the network graph for the placement of sample points. In previous work, candidates
for GPS measurements were usually road segments or nodes in the graph modeling the
network. In our case, candidate sets are limited to (or should prefer) station nodes in the
underlying network.
2.3.2 Map-Matching of Schedule Data
There is only little work on map-matching of schedule data. In [31], the authors describe
a greedy approach based on iterative shortest path calculations that is similar to our rst
baseline approach described in Section 2.4, but adds special auxiliary edges between node
candidates to avoid stalling. A global approach based on the construction of a “pseudo-
graph” was described in [122] and evaluated against the schedule data of Zurich on
an OpenStreetMap (OSM) network. The punishment of full turns and turn restrictions
2.4 Baseline Approaches 25
present in the OSM data as well as the usage of OSM attributes for better node candidate
selection where left as an open problem. In contrast, our work considers both types. A
preliminary version of our approach was published in [14].
2.3.3 Routing with Turn Restrictions
Our approach to handle turn restrictions is based on previous work in [149] and [71].
In [149], the concept of modeling turn restrictions by an edge-to-node dual graph was
studied. In [71], the goal was to improve both the time and space eciency of turn-
restricted shortest path searches in road networks. An edge-based Dijkstra’s algorithm
was described that is similar to our routing approach on the edge-to-node dual graph
(Section 2.6.4), and its applicability to Contraction Hierarchies was shown.
2.3.4 Curve Similarity
For our evaluation, we build on metrics for curve similarity and map-matching quality
previously described in the literature. A discrete variant of the Fréchet distance was
rst described in [53] together with an O(pq) algorithm for its computation, where p
and q are the number of discrete sample points in the compared curves P and Q. A
similarity metric between curves based on the average Fréchet distance is for example
described in [32] and was rst used as a quality metric for map-matching results in [30].
In [104], two additional metrics A
N
and A
L
based on the number and length of incorrectly
matched segments were introduced, of which we use A
N
in Section 2.10.
2.3.5 Similarity Classication of String Labels and POIs
The similarity measures used in our station similarity classication are related to pre-
vious work on string label similarity classication [29, 41, 78], often used for name dis-
ambiguation [57, 87]. Another area of research aims to nd pairs of points of interest
(POIs) describing the same real-world location [128, 129, 102, 46], which is for example
required to merge dierent geographical datasets without overlap.
2.4 Baseline Approaches
At rst glance, Problem 1 may seem like a classic routing problem: given an ordered list
of destinations, nd the optimal route which visits each destination in the given order.
This apparent similarity gives rise to several baseline approaches, which we will shortly
discuss in this section. We will gradually improve upon the simplest of these approaches
until we arrive at a basic hidden Markov model formulation in the next section.
26 Chapter 2. Map-Matching Schedule Data
Given a vehicle tripT of type λ which visits stations s
1
, ..., s
n
and a public transporta-
tion network N
λ
= (V , E)with stations S ⊆ V . Probably the simplest idea is to rst select
n candidate nodes v
1
, ..., v
n
,v
i
∈ S such that v
i
is the candidate node for s
i
, for example
by taking the v
i
closest to s
i
. Afterwards, search for the shortest paths between each
consecutive node candidates, with edge weights based on the travel time.
As there may be many tting station nodes for a single s
i
in S, an obvious extension
is to use candidate sets S
i
⊆ S for each station s
i
. To determine the path between an S
i
and its successors S
i+1
, we then calculate the set-to-set shortest path between S
i
and S
i+1
.
As described in Section 1.3.3, such a path can be found with a single run of Dijkstra’s
algorithm. However, when calculated separately, the shortest paths between individual
pairs S
i
and S
i+1
are not necessarily connected. We see three ways to tackle this problem:
node-to-set routing (NTS), local set-to-set routing (STS), and global set-to-set routing
(G-STS). Figure 2.5 gives examples for all three.
2.4.1 Node-To-Set Routing
Here, the paths between the candidate sets are not calculated individually. Instead, we
rst nd the shortest path from S
1
to S
2
. This settles a node v
2
∈ S
2
, namely the target
node of the found shortest path. For the path from S
2
, we then only search for the optimal
path from v
2
to S
3
. The resulting node v
3
is then again settled as the node for S
3
, and so
on.
2.4.2 Local Set-to-Set Routing
If instead the shortest paths between station candidate set pairs are calculated individu-
ally, the resulting “gaps” between the individuals shortest path may be closed either by
using a straight line segment (ignoring the network restrictions), or by calculating an
additional shortest path between the target node v
i
∈ S
i
belonging to the path from S
i−1
,
and the source node v
0
i
∈ S
i
belonging to the path to S
i+1
.
2.4.3 Global Set-to-Set Routing
This variant searches for the globally optimal path p through N that visits one node
of each candidate set S
1
, ..., S
n
in the given order. Such a path can be found by rst
calculating the shortest path between each pair (v, v
0
) ∈ S
i
×S
i+1
. Let P
i
denote the set
of paths for pairs S
i
×S
i+1
, and w.l.o.g. assume that these paths are given as node paths.
Formally, we then search for a sequence p
1
, ..., p
n−1
, p
i
∈ P
i
which minimizes
arg min
p
1
,... ,p
n−1
p
i
c(p
i
) (2.2)
2.5 Hidden Markov Model 27
under the constraint that the last node in p
i
is equal to the rst node in p
i+1
. This ap-
proach was used in [52]. The optimal path sequence can be found by building a directed
acyclic graph (DAG) in which the S
i
form node layers, it is hence often called a layered
graph. For each candidate set S
i
, all v ∈ S
i
are distinct nodes in this layer graph, and each
node for a v ∈ S
i
is connected with an edge to each node for a v
0
∈ S
i+1
, weighted by the
cost of the shortest path from v to v
0
, and labeled with a reference to the corresponding
path. Additionally, a special source node is connected to each v ∈ S
1
with an edge of
weight 0, and each v ∈ S
n
is connected to a special target node with an edge of weight
0. The shortest path on this DAG from the source to the target node is then the optimal
sequence p
1
, ..., p
n−1
.
2.5 Hidden Markov Model
Our baseline approaches so far ignored whether a station node candidate is the right
choice for a particular s
i
. In this section, we extend the G-STS approach by giving both
node candidates and the paths connecting them a probability score. Given a vehicle trip
to be map-matched, we then search for the most likely path the trip has taken through the
transportation network. This can be elegantly modeled using a hidden Markov model.
The hidden states H are the network’s station nodes (that is, H = S). In the re-
mainder of this chapter, we will use the terms hidden states and station candidates in-
terchangeably, denoted by h. The observations O are the stops Σ = s
1
, ..., s
N
visited by a
trip T in that order (that is, O = Σ).
2.5.1 Emissions
Previous work usually modeled the emission probability Pr(s | h) of observing s for
a location candidate h on the great-circle distance between s and h and assumed that
this distance followed a zero-mean normal distribution [86, 121, 109, 104]. The intuition
behind this is that the observed locations are prone to GPS measurement errors. It has
been acknowledged in previous work that GPS measurement errors do in fact not follow
a normal distribution [109].
We consider it a signicantly dierent problem to estimate the probability that a
GPS measurement matches a road segment than to estimate the probability that the
coordinate of a public transport vehicle stop matches a station in the transportation
network. Stop locations in schedule data are usually recorded manually, and there may
be many conicting philosophies of where to place the stop. Possible approaches are for
example the exact position on the tracks where the vehicle comes to a halt, the centroid of
the station area, the centroid of the station building, the entrance of the station building,
or the platform.
28 Chapter 2. Map-Matching Schedule Data
s
1
s
2
s
3
s
4
s
5
s
6
s
7
H
1
H
2
H
3
H
4
H
5
H
6
H
7
NTS
s
1
s
2
s
3
s
4
s
5
s
6
s
7
H
1
H
2
H
3
H
4
H
5
H
6
H
7
STS
s
1
s
2
s
3
s
4
s
5
s
6
s
7
H
1
H
2
H
3
H
4
H
5
H
6
H
7
G-STS
s
1
s
2
s
3
s
4
s
5
s
6
s
7
H
1
H
2
H
3
H
4
H
5
H
6
H
7
HMM
Figure 2.5: A transportation network N with a map-matching problem for a single ve-
hicle trip T , solved using the node-to-set baseline (NTS), the set-to-set baseline (STS),
the global set-to-set baseline (G-STS), and a hidden Markov model (HMM). The trans-
portation network is depicted in black. The network’s station nodes are depicted in red.
The trip’s station sequence Σ = s
1
, . . . , s
7
is depicted in dark green. All station nodes
within the search radius (light green) around an s
i
are used to build the set of station
candidates H
i
.
2.5 Hidden Markov Model 29
0 20 40 60 80 100 120 140 160 180 200
Distance in meters
0
2
4
×10
5
6
Number of similar pairs
Exponential distribution
Figure 2.6: Distance distribution between similar station pairs in ground truth data
obtained from OSM.
Based on an analysis [15] of stop positions belonging to the same station in ground
truth data we obtained from OSM, it is reasonable to assume that the distance between
these positions follows an exponential distribution (Figure 2.6). We therefore assume
that the geographical distance between the position π of an observation s and the posi-
tion Π(h) of a node candidate h is a random variable D with the density function
p(d; λ
e
) =
λ
e
e
−λ
e
d
if d ≥ 0,
0 else.
(2.3)
For a hidden state h and an observation s with distance d
s,h
, we then model the probability
Pr(s ≈ h) that h and s are a match as the probability that D is “big enough”:
Pr(s ≈ h) = Pr(D ≥ d
s,h
) = 1 − Pr(D ≤ d
s,h
) (2.4)
= 1 −
d
s, h
0
λ
e
e
−λ
e
d
(2.5)
= e
−λ
e
d
s, h
. (2.6)
We additionally normalize the Pr(s ≈ h) to make sure that they sum up to 1. Let
C
e
=
h∈H
s∈Σ
Pr(s ≈ h). (2.7)
We then model the emission probabilities as
Pr(s | h) =
1
C
e
Pr(s ≈ h) (2.8)
=
1
C
e
e
−λ
e
d
s, h
. (2.9)
30 Chapter 2. Map-Matching Schedule Data
ℎ
1
2
ℎ
1
1
ℎ
2
1
ℎ
2
2
ℎ
2
3
𝐻
1
𝐻
2
𝐻
3
Pr(ℎ
2
3
| ℎ
1
2
)
𝑞
0
Pr(ℎ
3
3
| ℎ
2
3
)
Pr(𝑠
1
| ℎ
1
)
Pr(𝑠
2
| ℎ
2
)
Pr(𝑠
3
| ℎ
3
)
ℎ
3
2
ℎ
3
3
ℎ
3
1
𝑠
1
𝑠
2
𝑠
3
1
1
Figure 2.7: Hidden Markov model for a map-matching problem of a trip T with 3 stops
s
1
, s
2
, and s
3
. Dashed arrows denote transitions, solid arrows denote emissions. Transi-
tion and emissions with zero probabilities are omitted.
Observe that C
e
≥
h∈H
e
−λ
e
d
for a single s and hence
h∈H
Pr(s | h) =
1
C
e
h∈H
e
−λ
e
d
≤
1. For each h, the remaining probability weight 1 −
1
C
e
h∈H
e
−λ
e
d
is put into an emission
of a dummy nil observation . We will see in the following section that this normal-
ization factor has no inuence on the optimal sequence of hidden states through the
HMM.
For most pairs (s, h), Pr(s | h) will be very close to zero, as the geographical dis-
tance between them is very big. It is therefore reasonable to only calculate the emission
probabilities for pairs that are geographically close (within a threshold distance r ) and
assume the remaining emission probabilities to be 0. For a given stop s
i
∈ Σ, we will use
the following notation throughout the rest of this chapter: H
i
⊆ H is the set of potential
stop candidates within a distance r of s
i
. We will sometimes denote by h
i
that a stop
node is a candidate for s
i
, that is h
i
∈ H
i
. In general,
H
i
, H (there might be network
stations that are not within the threshold distance for any stop s) and H
i
∩ H
j
, ∅ for
i , j (stops might share the same candidate stations).
2.5.2 Transitions
We base the transition probability between two hidden states h and h
0
on the travel time
t of the shortest path between them. Based on the intuition that a public transit vehicle
will most likely travel to the next station candidate that is closest in terms of travel time,
we model t as a random variable T following an exponential distribution. For a given t
2.5 Hidden Markov Model 31
and a small ϵ > 0, the probability that T is within a range ϵ around t is then
Pr(|T −t | < ϵ) ≈ ϵλ
t
e
−λ
t
t
. (2.10)
As with the emission probabilities in the previous section, we cannot use Pr(|T −t | < ϵ)
directly as the transition probability. To transform this continuous probability distribu-
tion over travel times into a discrete distribution over transitions from some hidden state
h to a some hidden state h
0
, we dene
C
t
=
h∈H
h
0
∈H
Pr(|T −t
h,h
0
| < ϵ) (2.11)
as the sum over all Pr(|T −t
h,h
0
| < ϵ), (h, h
0
) ∈ H ×H , where t
h,h
0
is the travel time of the
shortest path between h and h
0
. We then model the transition probabilities as
Pr(h
0
| h) =
1
C
t
Pr(|T −t
h,h
0
| < ϵ) (2.12)
=
1
C
0
t
e
−λ
t
t
h, h
0
. (2.13)
The remaining probability weight for each hidden state is put into a transition to itself.
The normalization by C
e
and C
0
t
is strictly for conformity with the denition of a
hidden Markov model and has no inuence on the optimal sequence of hidden states,
which will become clear in the following. For a given observation sequence s
1
, ..., s
N
, we
now want to nd the hidden state sequence i
1
, ..., i
N
with maximum probability:
arg max
i
1
,... ,i
N
Pr(i
1
, . . . , i
N
| s
1
, . . . , s
N
) (2.14)
=arg max
i
1
,... ,i
N
N
k=1
Pr(i
k
| i
k−1
) · Pr(s
k
| i
k
) (2.15)
=arg max
i
1
,... ,i
N
1
C
0
t
C
e
N
k=1
Pr
T −t
i
k−1
,i
k
< ϵ
· Pr(s
k
≈ i
k
) (2.16)
=arg max
i
1
,... ,i
N
N
k=1
Pr
T −t
i
k−1
,i
k
< ϵ
· Pr(s
k
≈ i
k
). (2.17)
In the literature on map-matching using a HMM the transition and emission proba-
bilities are typically given directly, without the normalization factor, although this does
not strictly t the HMM denition.
32 Chapter 2. Map-Matching Schedule Data
2.5.3 Transformation to Log Probabilities
As the probabilities tend to get very small, special care has to be taken to avoid nu-
merical issues, in particular oating point underow. A standard method is to optimize
Pr(i
1
, . . . , i
N
| s
1
, . . . , s
N
) in log space:
arg max
i
1
,... ,i
N
Pr(i
1
, . . . , i
N
| s
1
, . . . , s
N
) (2.18)
=arg max
i
1
,... ,i
N
ln Pr(i
1
, . . . , i
N
| s
1
, . . . , s
N
) (2.19)
=arg max
i
1
,... ,i
N
ln
N
k=1
Pr
T −t
i
k−1
,i
k
< ϵ
· Pr(s
k
≈ i
k
)
(2.20)
=arg max
i
1
,... ,i
N
N
k=1
ln Pr
T −t
i
k−1
,i
k
< ϵ
+ ln Pr(s
k
≈ i
k
). (2.21)
By dropping additive constants, we get
arg max
i
1
,... ,i
N
Pr(i
1
, . . . , i
N
| s
1
, . . . , s
N
) (2.22)
=arg max
i
1
,... ,i
N
N
k=1
−λ
t
t
i
k−1
,i
k
−λ
e
d
s
k
,i
k
(2.23)
=arg min
i
1
,... ,i
N
N
k=1
λ
t
t
i
k−1
,i
k
+ λ
e
d
s
k
,i
k
. (2.24)
The scale factors λ
t
and λ
e
are parameters of our approach.
2.5.4 Optimal Hidden State Sequence as a Shortest Path Problem
From Equation 2.24 it immediately follows that we can nd the optimal hidden state
sequence i
1
, . . . , i
N
by calculating a shortest path in a special directed acyclic graph U =
(V
U
, E
U
). For each hidden state h
i
∈ H
i
we add a node to U . Additionally, we add a node
for the initial state q
0
and a special sink node t. t is connected to each h
N
by an edge
with weight 0. q
0
is connected to each h
1
by an edge with weight λ
e
d
s
1
,h
1
. The remaining
nodes h
i
with 1 < i < N are connected to each node h
i+1
with an edge with weight
w(e) = λ
e
d
s
i+1
,h
i+1
+ λ
t
t
h
i
,h
i+1
. Figure 2.8 gives an example.
It is easy to see that the node sequence of the shortest path from q
0
to t will then op-
timize Equation 2.24. As U is a directed acyclic graph, the shortest path can be found in
O(|V
U
|+ |E
U
|) by rst obtaining a topological ordering of the V
U
, and then by processing
them in that order, relaxing their adjacent neighbors. The key insight is that because of
the topological ordering, if we process a node v, the cost label of v is already optimal
2.6 Turn Costs 33
ℎ
1
2
ℎ
1
1
ℎ
2
1
ℎ
2
2
ℎ
2
3
𝐻
1
𝐻
2
𝐻
3
𝑞
0
𝑡
ℎ
3
2
ℎ
3
3
ℎ
3
1
𝜆
𝑒
𝑑
𝑠
1
,ℎ
1
1
0
0
𝜆
𝑒
𝑑
𝑠
1
,ℎ
1
2
𝜆
𝑡
𝑡
ℎ
1
2
,ℎ
2
3
+ 𝜆
𝑒
𝑑
𝑠
2
,ℎ
2
3
𝜆
𝑡
𝑡
ℎ
2
3
,ℎ
3
3
+ 𝜆
𝑒
𝑑
𝑠
3
,ℎ
3
3
0
Figure 2.8: Finding the optimal hidden state sequence through the hidden Markov
model from Figure 2.7 as a shortest path problem. Edge weights are depicted in black for
4 selected edges. The node sequence of the shortest path from source q
0
to sink t (without
nodes q
0
and t) exactly corresponds to the optimal hidden state sequence i
1
, . . . , i
N
.
(there cannot be a better path to v because we have already processed all possible pre-
decessor nodes of v). Even better, because of the way we construct the HMM graph, we
already have a topological ordering. Note that |V
U
| ∈ O(|H |·|O |)and |E
U
| ∈ O(|O|·|H |
2
).
The asymptotic running time can thus also be expressed as O(|O |·|H |
2
), which is equiv-
alent to the asymptotic running time of the Viterbi algorithm mentioned above.
2.6 Turn Costs
A standard issue in road network routing is handling turn restrictions. Consider Fig-
ure 2.9, left: a vehicle coming from some street e may not be allowed to enter into street
f , whereas a vehicle coming from street h is allowed to enter. More generally, turning
from one edge into another might not be restricted per se, but produce some cost:
Denition 2.7 (Turn Cost Function). Given a directed graph G = (V , E) and the set
A ⊆ E × E of adjacent edges (we say (u, v) and (v, w) are adjacent, but not (u, v) and
(w,v)). A turn cost function is a function R : A 7→ R
+
0
that assigns two adjacent edges e
and f a turn cost for traveling from e to f .
In the context of schedule map-matching, where sample points are typically very
sparse, turn costs and turn restrictions have to be included into the shortest path calcu-
lations for the individual HMM transitions to ensure correct paths.
34 Chapter 2. Map-Matching Schedule Data
𝐴
𝐵
𝐶
𝐷
𝐸
𝑓
𝑒
𝑔
ℎ
𝑖
𝐴
𝐵
𝐶
𝐷
𝐸
𝑒
𝑓
𝑔
ℎ
𝑖
𝑎
𝐴
𝐵
𝐶
𝐷
𝐸
Figure 2.9: Left: Coming from node B, there is a turn restriction at A that prohibits
turning into segment f . Middle: A typical approach to model this is to blow up the node
A to get explicit edges for each turn. In this example, the weight of edge a can now hold
the turn cost. Right: If nodes are adjacent to many two-way street segments, the number
of required edges quickly blows up
An often-used approach to model turn costs in directed graphs is shown in Figure 2.9,
middle: aected nodes are expanded and “entry” nodes for each adjacent edge are added.
Afterwards, edges for each possible turn are added, and may then be given weights
corresponding to the turn costs. A drawback of this approach is that it may add O(D)
additional entry nodes, and O(D
2
) additional turn edges to the graph, where D is the
maximum sum of the in- and out-degree for any node in the graph. For example, if n is a
node in a street network with 4 adjacent roads which can each be used in both directions,
we must expand n into 4 entry nodes, and add 3 edges to each entry node to cover all
possible turns. Figure 2.9, right, gives an example. A rst idea to slightly mitigate this is
to contract the non-turn edges (for example, д, h, e, and f in Figure 2.9, middle). In this
example, this leaves single nodes representing the original edges д, h, e, and f , namely
D, E, B, and C. It is not immediately clear how to perform this contraction without losing
information required for nding correct shortest paths later on, though.
This section rst describes two basic types of turn costs that are relevant for schedule
map-matching of road and rail bound public transit vehicles: mandatory turn restrictions
and angular turn costs. We then continue to describe the problem of inter-hop turn costs
(turn costs between hidden state transitions). The remainder of this section then extends
and formalizes the edge contraction idea from the last paragraph and explains how turn
costs can be elegantly modeled using the edge-to-node dual of the transportation network
N. In particular, we prove that no shortest paths are lost in this transformation and
describe how to preserve all original edge weights in the process. We also show how
inter-hop turn restriction can be naturally considered with this approach.
2.6 Turn Costs 35
𝑡
𝑠
𝑣
𝑒
𝑓
Figure 2.10: The shortest path from s to t is shown in blue - but a rail-bound vehicle is
physically unable to make the sharp turn from edge e to edge f at node v. To make the
lower path more likely, angular turn costs have to be considered.
2.6.1 Mandatory Turn Restrictions
Mandatory turn restrictions are regulated by trac laws or (additionally) by physical
barriers, for example a green corridor between road lanes. It is usually not possible to
circumvent them, and it is therefore reasonable to model them as innite turn costs.
Denition 2.8 (Turn Restrictor). We call a turn cost function R
∞
7→ {0, ∞} a turn
restrictor.
2.6.2 Angular Turn Costs
Depending on the vehicle type, we also have to consider turn restrictions stemming
from physical limitations of the vehicle. Consider the example given in Figure 2.10. It is
usually very time-consuming for a train to make a full turn on open track, although it
would theoretically be possible (the train might stop, the driver could change to the rear
cabin and continue in the opposite direction). Paths that use acute angles like the one
in Figure 2.10 are immediately recognizable by humans as unrealistic. However, if we
assume the edge weights in Figure 2.10 to be the travel time based on a uniform vehicle
velocity, the path highlighted in blue is the optimal path from s to t. Similarly, it may
take a bus some time to make an acute turn at an intersection, or to make a U-turn at a
roundabout.
To consider this during map-matching, we add an angular turn cost function R
ϕ
which is directly based on the angle between two adjacent edges e = (u, v) and f =
(v, w). As N is embedded in R
2
by Π, this angle is given by ϕ(e, f ) = arccos(
ˆ
a ·
ˆ
b), where
a = Π(u) − Π(v) and b = Π(w) − Π(v).
2.6.3 Inter-Hop Turn Restrictions
During our map-matching process, we also want turn restrictions to be considered be-
tween transitions. We call these restrictions inter-hop turn restrictions. Consequentially,
36 Chapter 2. Map-Matching Schedule Data
ℎ
0
0
ℎ
0
2
ℎ
0
1
ℎ
0
0
ℎ
0
2
ℎ
0
1
Figure 2.11: Shortest-path transitions between three hidden states h
0
0
, h
0
1
, and h
0
2
. Left:
Turn restrictions are not considered between hops, leading to a full U turn if both short-
est path transitions are combined. Right: Turn restrictions are considered, leading to a
smooth overall path.
A
B
C
D
E
e
f
д
h
i
p
f
g
i
h
e
p
Figure 2.12: Left: A primal digraph G (black). Right: Its edge-to-node dual D(G) (red).
The blue path p from f to e is simple in D(G), but its primal representation p from B to
C in G is not (node A is visited two times).
turn restrictions that should be considered during a transition are sometimes called intra-
hop turn restrictions. As the hidden states in our HMM are station nodes in the trans-
portation network, and as a HMM is memoryless, it is not immediately clear how this
can be ensured. Consider Figure 2.11, left. Two transition paths from hidden state h
0
0
to
hidden state h
0
1
, and from h
0
1
to h
0
2
are shown, as calculated by one shortest path search
from h
0
0
to h
0
1
, and one from h
0
1
to h
0
2
. At the point of calculating the path from h
0
1
to h
0
2
, we
have “forgotten” the direction from which we arrived at h
0
1
, and the calculated shortest
path to h
0
2
will introduce a sharp U-turn at h
0
1
. We would like our turn cost function to
also take eect here. To achieve this, we have to encode the current vehicle heading into
the hidden states. The basic idea, laid out in detail in the following section, is to not take
nodes as hidden states, but edges.
2.6 Turn Costs 37
2.6.4 Edge-to-Node Dual
The edge-to-node dual D(G) of a graph G (then called the primal graph), rst described
in [147], is constructed by adding nodes for each edge of G, and by connecting two
dual nodes if their primal edges were adjacent. Figure 2.12 gives an example. The edge-
to-node dual is also often called line graph, but this term is already used for another
concept in this work, so we use this more explicit name. Formally, the edge-to-node
dual is dened as follows:
Denition 2.9 (Edge-to-Node Dual). Given a primal graph G = (V , E), we say D(G) =
(V, E) is its edge-to-node dual if the following holds:
1. There is a bijection d
v
: E 7→ V (each primal edge has a distinct dual node).
2. Let E
2
be the set of adjacent edges in G (if G is directed, we say f is adjacent to e
if head(e) = tail(f ), otherwise we say f and д are adjacent if they share a node).
There is a bijection d
e
: E
2
7→ E such that d
e
(e, f ) = (d
v
(e), d
v
(f )) (if two primal
edges were adjacent, their dual nodes are adjacent).
We rst prove several interesting properties which will help to clarify its applicability
to the problem of turn restrictions.
Lemma 2.1. For each simple primal path p with |p| > 0 through G there is a unique simple
dual path p through D(G).
Proof. As p is simple and |p| > 0 the path’s edge sequence e
1
, . . . , e
n−1
implies a unique
sequence of pairwise distinct nodes d
v
(e
1
), . . . , d
v
(e
n−1
) = v
1
, . . . , v
n−1
. Since each e
i+1
is
adjacent to e
i
, v
i+1
is adjacent to v
i
by edge d
e
(e
i
, e
i+1
). This dual node sequence therefore
is a path p through D(G). As the sequence was unique for p, so is p.
We can therefore represent each nonempty simple path in G by exactly one distinct
simple path in D(G). There are no nonempty simple paths “lost” in the edge-to-node
dual. An important consequence of this and Lemma 1.1 is that each nontrivial shortest
path through G with edge weights > 0 has a unique dual representation.
Lemma 2.2. For each simple dual path p in D(G) there is a unique primal path p.
Proof. The path’s node sequence v
1
, . . . , v
n
implies a unique edge sequence through G
d
v
−1
(v
1
), . . . , d
−1
v
(v
n
). By Denition 2.9, this edge sequence is a path.
Lemma 2.3. This path p may not necessarily be simple.
Proof. Consider the counter-example given in Figure 2.12.
38 Chapter 2. Map-Matching Schedule Data
v
u
w
q
e
1
e
3
e
2
v
u
w
q
Figure 2.13: Left: The state of a vehicle being at node v is encoded by v itself, without
any information regarding the direction it is headed to. Right: The state of a vehicle
being at node v is encoded by all its outgoing edges. For example, e
3
encodes the state
of being at v, heading in the direction of u.
Observe that |p| = |p| + 1 in general. To compute routes in the edge-to-node dual,
we now search for shortest paths between nodes that correspond to primal edges and
thus to segments in the underlying transportation network. Two questions immediately
arise: (1) If we want to search for a shortest path between primal nodes s and t, what is
a corresponding shortest path query in the edge-to-node dual? (2) How should the dual
edge weights be modeled?
Candidate Edges
We now assume that the primal graph G is directed. As primal nodes have no immediate
dual representation, we have to nd a way to encode a primal shortest path query in the
primal graph in a shortest path query in the edge-to-node dual. To this end, we represent
a primal node v by a set ∆(v) of gateway edges. ∆(v) consists of all outgoing edges from
v. For simplicity, we denote by d
v
(∆(v)) the set of dual nodes corresponding to ∆(v).
Intuitively, an edge e = (v, u) ∈ ∆(v) and its corresponding dual node d
v
(e) encode
the state of being at node v, heading towards u. Figure 2.13 gives an example.
To calculate the shortest path from primal nodes s to t in the edge-to-node dual, we
now have to nd the shortest path between the sets d
v
(∆(s)) and d
v
(∆(t)). This is again a
standard set-to-set shortest path problem and can be computed by Dijkstra’s algorithm
as described in Section 1.3.3.
Dual Edge Weights
Special care has to be taken to ensure that primal edge costs are correctly reected in
the edge-to-node dual. In particular, if no turn costs are considered, we want the cost
for a shortest primal path from s to t to be equivalent to the cost of the corresponding
dual shortest path from d
v
(∆(s)) and d
v
(∆(t)).
2.6 Turn Costs 39
Given a weighted graph G with a weight function w, we dene the corresponding
weight function in the edge-to-node dual as
w(e) = w
d
−1
v
(
tail
(
e
))
. (2.25)
In words: the weight of a dual edge e = (u, v) is the weight of the primal edge corre-
sponding to its tail node.
Lemma 2.4. Let p be a path in D(G) with edge sequence e
1
, . . . , e
|p|
and ending at node
t. Its corresponding primal path is p with e dge sequence e
1
, . . . , e
|p|+1
. Let p
0
the path p
extended by any node t
0
∈ adj
+
(t). Let G be weighted by w and let D(G) be weighted
according to Equation 2.25. Then c(p) = c(p
0
).
Proof.
c(p
0
) =
|p|+1
i=1
w
d
−1
v
(
tail
(
e
i
))
(2.26)
= w(e
1
) + ··· + w(e
|p|+1
) (2.27)
=
|p|
i=1
w(e
i
) = c(p). (2.28)
Lemma 2.5. Let G be a primal graph weighted by w > 0 and its edge-to-node dual D(G) be
weighted by w according to Equation 2.25. The shortest dual path from d
v
(∆(s)) to d
v
(∆(t))
describes a shortest simple primal path from s to t.
Proof. Let p
0
be such a path with edge sequence e
1
, . . . , e
n+1
and cost c(p
0
). Observe that
tail(d
−1
v
(e
1
)) = s and tail(d
−1
v
(e
n+1
)) = head(d
−1
v
(e
n
)) = t. Let p be the path p
0
with the last
edge removed. According to Lemma 2.2, p describes a unique path p in G, and this will
start at s and end at t. It remains to show that p is really a shortest simple path from s to t.
Assume that is not the case and that there is a primal path q from s to t with c(q) < c(p).
According to Lemma 2.1, this new path q also has a unique dual representation q. As q
still begins at s, q still begins at a node in d
v
(∆(s)). As q still ends at t, q still ends at a node
adjacent to d
v
(∆(t)). We can thus extend q by a single dual node into a path q
0
which ends
in d
v
(∆(t)). But according to Lemma 2.4, c(q
0
) = c(q), and therefore c(q
0
) < c(p
0
), which
means p
0
was not the shortest dual path, a contradiction. p therefore was a shortest path
from s to t. As w > 0, it was also simple.
Figure 2.14 gives an example of a shortest path problem in a primal graph and the
corresponding shortest path problem in the edge-to-node dual.
40 Chapter 2. Map-Matching Schedule Data
E
C
A
B
D
1
6
2
5
3
3
6
1
3
3
2
5
6
e
f
e
f
g
д
Figure 2.14: Left: A shortest path problem from A (red) to D (green) in a weighted primal
graph G. Numerical labels depict the edge weights. Right: The corresponding shortest
path problem in its edge-to-node dual. Instead of single source and target nodes, we now
have a set of dual source nodes {e, f} (red) corresponding to primal edges (light red) and
a set of dual target nodes {g} (green) corresponding to primal edges (light green).
E
C
A
B
D
1
6
2
5
3
3
6
1
3
3
2
5
6
e
f
e
f
F
9
3
9
v
Ω
v
∞
Figure 2.15: Left: A shortest path problem from A (red) to D (green) in a weighted
primal graph G. Node F has no outgoing edge. To ensure a non-empty target node set
in the edge-to-node dual, we add a virtual node Ω and a virtual edge v = (F, Ω). Right:
The corresponding dual shortest path problem.
We have so far assumed that the sets ∆(s) and ∆(t) are nonempty, which may not be
the case. For ∆(s), this is irrelevant, as we cannot nd a shortest path from a node s to
any other node if s has no outgoing edge. For t, however, a lack of outgoing primal edges
means we cannot arrive at node t in the edge-to-node dual. Consider Figure 2.15. To be
able to calculate a shortest path from A to F in the edge-to-node dual, we temporarily
insert a virtual node Ω and a virtual edge v = (F , Ω). The weight w(v) is set to innity.
As Ω will have no outgoing edge, w(v) will not appear in any dual edge weight, but we
are now able to calculate a shortest path to F (Figure 2.15, right).
Turn Costs in the Edge-To-Node Dual
In the edge-to-node dual, the consideration of angular, mandatory, and inter-hop turn
costs is now straightforward. Given a turn cost function R, we may simply update the
dual weight function from Equation 2.25 to
w(e) = w
d
−1
v
(
tail
(
e
))
+ R
d
−1
v
(
head
(
e
))
, d
−1
v
(
tail
(
e
))
. (2.29)
2.7 Incorporating Network Meta Data 41
A
1
6
2
5
3
3
6
1
3 + ∞
3 + 10
2
5
6 + 3
e
f
h
e
f
д
∞
10
3
h
g
i
i
Figure 2.16: Left: A weighted graph G with a turn cost function R. Non-zero turn costs
between edges are depicted in red, for example R(e, f ) = 3. Right: The corresponding
dual shortest path problem, turn costs are added to the corresponding dual edge.
See Figure 2.16 for an example.
To nally be able to consider inter-hop turn restrictions as explained above, we
slightly change the denition of our hidden Markov model from Section 2.2.2. As hidden
states H , we now no longer consider network station nodes v ∈ S, but their outgoing
edges: H =
v∈S
∆(v). As explained above, a hidden state now no longer encodes being
orientationless at a station, but being at a station and heading in a particular direction.
A subtle point has to be addressed before we can close this section. As we are choos-
ing the outgoing edges ∆ (v) as new hidden states for a network station v ∈ S, a turn
restriction for entering an edge e ∈ ∆(v) might be added to the path cost already when
we arrive at v, although the vehicle has not yet entered the network segment corre-
sponding to e. If v is not the terminus for a map-matched trip, this is desired, as we have
to leave through some outgoing edge anyway for the next hop, at which point this turn
cost would actually apply. For a terminus station, however, this is not the case.
A simple solution to this problem is to add virtual nodes and edges like described
above for all terminus station candidates, with no attached turn restrictions.
2.7 Incorporating Network Meta Data
Real-world datasets for public transportation networks often come with additional meta-
data that goes beyond the basic representation of network segments. This includes for
example mandatory turn restrictions as already described, one-way streets, or speed lim-
its. While the details of how such data can be transformed into a routable transportation
network N would be out of scope here, there are two kinds of relevant metadata specic
to our problem: public transportation routes, and station labels. This section will briey
describe how to incorporate them into our approach. We will focus on the largest dataset
for transportation networks, OSM, which is also used in our experimental evaluation.
42 Chapter 2. Map-Matching Schedule Data
2.7.1 Public Transit Route Relations
OSM already contains information on public transportation routes via the relation route
5
.
For example, relations of type route=train group network segments which belong to
the route of a single train line. This line is specied via an attribute line=* which gives
the line name (e.g. “Intercity-Express Basel SBB -> Hamburg Altona”) and/or a ref=*
attribute which gives the train number (e.g. “ICE 445”). Similar relations exist for other
public transport vehicle types, for example streetcars, busses, or ferries.
At rst glance, it may seem sucient to extract a matching route relation for a ve-
hicle trip we want to map-match. There are several reasons why this approach does not
work: (1) The geographical coverage of route relations is not good enough, especially in
rural areas. (2) The schedule coverage is usually not good enough. In particular, outlier
trips (vehicles returning to the depot, additional irregular services during peak hours,
etc.) are often missing. (3) The relations are often outdated. With each new schedule,
route relations have to be updated manually. As these relations do not prominently
appear in the standard OSM map tiles, outdated routes often remain unnoticed. (4) The
relations itself are sometimes erroneous or have holes.
Despite these issues, route relations provide valuable information on the vehicle
path. To incorporate this, we label each edge contained in a public transit route relation
with the attribute value line=* and ref=*. Let now L(e) be the set of line labels for an
edge e in our transportation network, taken from the OSM attributes, and let l be the
line label of a trip T . Instead of static edge weights w(e) based on the traversal time we
now use an edge weight function f : E × T 7→ R
+
:
f (e,T ) =
w(e) if l ∈ L(e),
w(e) ·w
L
else.
(2.30)
Here, w
L
≥ 1 is a penalty factor for edges with no matching lines.
2.7.2 Station Labels
Similarly, we would like to prefer network station candidates with a label that matches
the station label in the schedule data. To this end, we add the station label attributes name
and uic_name to the label set N (v) of the corresponding network node v. To prefer net-
work station node candidates with matching labels, we update our emission probabilities
and now calculate the probability Pr(s ≈ h) that a schedule stop s matches a network
station node candidate h as follows:
5
https://wiki.openstreetmap.org/wiki/Public_transport#Service_routes
2.8 Advanced Station Similarity Classication 43
Hauptbahnhof
Freiburg
Freiburg Hauptbahnhof
Freiburg
Hbf
Freiburg im Breisgau
Hauptbahnhof Freiburg
Freiburg im Breisgau Hauptbahnhof
Freiburg (Breisgau) Hauptbahnhof
Hauptbahnhof
,
Freiburg im Breisgau
Freiburg (Breisgau)
,
Hauptbahnhof
Freiburg (Brsg) Hauptbahnhof
Freiburg (Brsg)
Hbf
Figure 2.17: Some labeling variants for Freiburg Main Station. Same colors symbolize
semantically equivalent tokens.
Pr(s ≈ h) =
Pr(D ≥ d
s,h
) if n ∈ N (h),
Pr(D ≥ d
s,h
) ·w
N
else,
(2.31)
where n is the label of scheduled stop s, and w
N
≥ 1 is a penalty factor.
2.8 Advanced Station Similarity Classication
A problem with the updated emission probabilities from above is that stations are typ-
ically not labeled consistently in real-world datasets. A hard equivalency check would
penalize a station node candidate labeled “London St Pancras” for a scheduled stop la-
beled “London, St Pancras”, although they might both have the exact same geographical
position. Clearly this is not desired. Instead, we would like to have a robust classier
to decide whether two stations are similar. We dene such a station identier z as a
triple (n, ϕ, λ). n is the station name, ϕ its geographical latitude, and λ its geographical
longitude. This section then studies the following problem:
Problem 2 (Station Similarity Classication). Find a classication function c that maps
pairs (z
a
, z
b
) of station identiers to {0, 1}. We want c(z
a
, z
b
) = 1 if z
a
and z
b
describe the
same real-world station, and c(z
a
, z
b
) = 0 otherwise.
This section describes the characteristics of station identiers as they typically ap-
pear in real-world datasets. We then discuss several approaches to Problem 2. All ap-
proaches will be evaluated in Section 2.10.
2.8.1 Characteristics of Station Identiers
Both the geographic coordinates and labels between similar station identiers may dif-
fer signicantly. Figure 2.17 gives an (incomplete) list of labeling variants for Freiburg
Hauptbahnhof (the main station in Freiburg, Germany) as they appear in OSM and
44 Chapter 2. Map-Matching Schedule Data
London King's Cross
London King's Cross
King's Cross Station
King's Cross Station
London St.Pancras
London St.Pancras
London St. Pancras International
St Pancras International Station
St Pancras
London St Pancras
London St Pancras
London St Pancras
St Pancras
St Pancras
London St Pancras International
London St. Pancras International
International Station
5
0
m
1
0
0
m
1
5
0
m
2
0
0
m
2
5
0
m
Figure 2.18: Station identiers for London St Pancras in three dierent datasets: Open-
StreetMap ( OSM, black), Deutsche Bahn schedule ( DB, green), Association of Train
Operating Companies schedule ( ATOC, red), EuroStar schedule ( ES, blue). The DB
station has a distance of over 200 meters to the ES stations. For both ES stations, the
nearest station in the OSM dataset is King’s Cross station.
schedule data for the local transportation authority. Similarly, Figure 2.18 shows sev-
eral labeling variants for London St Pancras. In these examples, many of the typical
characteristics of station identier labels in western languages are present:
Few typos. Typos in station labels are rare. Station labels are very short and often
displayed prominently, where typos would be quickly noticed.
Abbreviations. Station labels often contain abbreviations. These are either specic
to a public transportation network, regionally specic (like “Freiburg (Brsg)” for
“Freiburg im Breisgau”), typical for entire countries (like “Hbf” or “HB” for ”Haup-
bahnhof“ in Germany and Switzerland, respectively) or language specic (like “Str.”
or “St.” for “street” in German and English, respectively). Often the abbreviations
are completely regional and hard to decipher even for locals (like “BBrunnen” for
“Bertoldsbrunnen” in Freiburg).
Varying token separators. Tokens may be separated by space, varying punctuation
(commas, semicolons, hyphens, brackets), or camel casing, among others.
Omitted location sp eciers. Tokens which specify a geographic location (countries,
cities, districts etc.) may be omitted either completely or partially, either because
they are clear from context, or because the data is regional and just includes a single
city.
2.8 Advanced Station Similarity Classication 45
Omitted station spe ciers. Tokens which exactly specify a station may be omitted.
This is often the case in long-distance schedule data, where trains only make a
single stop per city, and where cities only have a single major railway station.
Dierent token ordering. Token ordering is often arbitrary. For example, “Freiburg
Hauptbahnhof” and “Hauptbahnhof Freiburg” are equivalent.
There may also be considerable dierences between the coordinates of similar station
identiers. Figure 2.18 depicts the position of several identiers for London St Pancras
station in OSM, the ocial Deutsche Bahn schedule, the ocial schedule of the Asso-
ciation of Train Operating Companies (ATOC), and the schedule data for the EuroStar
lines. Again, several of the typical characteristics of station identier coordinates are
present:
Precision issues. Coordinates are often imprecise, e.g. because of truncation, or be-
cause they were manually compiled by clicking on a map.
Dierent placement philosophies. Station identiers might be placed at the station
entrance (there might be more than one), at the centroid of the station building
(there might also be more than one building), at the centroid of the station polygon
(the latter is not well-dened), at the centroid of the platform the vehicle arrives at,
at the exact position the vehicle will come to a halt, at the exact position the center
of the vehicle will on average come to a halt, or at a position that is best suited to
render a station label.
Human error. The location of a station identier might simply be wrong because of
human error. For example, a station identier coordinate might have been erro-
neously assigned to a neighboring station.
2.8.2 Classication Techniques
In order to tackle these characteristics, this section describers several techniques for sta-
tion similarity classication. We rst describe some naive baseline approaches which
will later be also evaluated for comparison. Afterwards, some techniques for label sim-
ilarity based on established measures for string similarity are discussed. We close with
feature engineering techniques to make the problem accessible to machine learning.
Position Equivalency
A naive classier would just check the positions of the station identiers for equivalency,
using some ϵ to account for oating point inaccuracies.
c
PEQ
(z
a
, z
b
) =
1 if dist(ϕ
a
, λ
a
, ϕ
b
, λ
b
) < ϵ,
0 else.
(2.32)
It is trivial to construct examples where c
PEQ
would produce false negatives.
46 Chapter 2. Map-Matching Schedule Data
Label Equivalency
In Section 2.7.2, we implicitly used a classier based on label equivalency:
c
LEQ
(z
a
, z
b
) =
1 if n
a
= n
b
,
0 else.
(2.33)
As described above, such a classier would lead to many false negatives if used in a
standalone fashion, but also to many false positives, as dierent stations in dierent
cities might share the same label.
Label Normalization
Dierences in spelling, common abbreviations, problems with lower- and uppercase,
and redundant tokens (e.g. a station label suxed with "station") might be normalized
by manually created normalization rules.
Position Similarity
We now turn to classication based on label similarity measures. In the context of this
work, a similarity measure is a function that maps to [0, 1] and quanties the similarity
between its arguments (a higher value means higher similarity). To use these similarity
measures for classication, a threshold value is dened.
The most basic similarity measure between station identiers is based on the geo-
graphical distance. We recall Figure 2.6 and note that the geographical distance between
matching station identiers in real-world data seems to t an exponential distribution.
Correspondingly, we model the position similarity sim
POS
as follows:
sim
POS
= exp
(
−λ · dist
(
ϕ
1
, λ
1
, ϕ
2
, λ
2
))
(2.34)
We would like to have sim
POS
(z
a
, z
b
) = 0.5 if z
a
and z
b
have a distance
ˆ
d (the threshold
distance) and therefore set λ = (ln 2)/
ˆ
d.
Edit Distance Similarity
A classic string metric is the edit distance, or Levenshtein distance. It gives the minimum
number of edit operations (either insert, deletion or substitution) necessary to transform
a string a into a string b. The edit distance is symmetric, i.e. ed(a, b) = ed(b, a). An often
used similarity measure based on the edit distance uses the ratio between ed(a, b) and
2.8 Advanced Station Similarity Classication 47
the length of the longer string:
sim
ED
(z
a
, z
b
) = 1 −
ed(n
a
, n
b
)
max(|n
a
|, |n
b
|)
. (2.35)
It is again easy to construct examples where this measure underestimates similarity:
with n
a
= “Freiburg im Breisgau, Hauptbahnhof” and n
b
= “Hauptbahnhof, Freiburg im
Breisgau”, sim
ED
(z
a
, z
b
) = 0.17, although they clearly describe the same station.
Prex and Sux Edit Distance
With n
a
= “St Pancras” and n
b
= “St Pancras International Station”, sim
ED
(z
a
, z
b
) =
0.31. To get a similarity measure that is more robust against missing tokens, we could
use the prex or sux edit distance. The prex edit distance (PED) is dened as
ped(a, b) = min
b
0
ed(a, b
0
). (2.36)
where b
0
is a prex of b. Similarly, the sux edit distance SED may be dened. In this
case, b
0
is then a sux of b. Neither the PED nor the SED are symmetric. To still get a
symmetric similarity measure, we compute both directions and take the best result:
sim
PED
(z
a
, z
b
) = 1 − min
ped
(
n
a
, n
b
)
|n
a
|
,
ped
(
n
b
, n
a
)
|n
b
|
. (2.37)
Then, again with n
a
= “St Pancras” and n
b
= “St Pancras International Station”,
sim
PED
(z
a
, z
b
) = sim
PED
(z
b
, z
a
) = 1. A similarity measure for the SED can be analogously
dened. It is still easy to construct examples where this measure overestimates similarity.
For example, the station labels “Berlin Zoo” and “Berlin Gesundbrunnen” have a PED
similarity of 0.7, but describe two dierent railway stations in Berlin.
Jaro Similarity
The Jaro Similarity [89] is especially well suited for comparison of short strings. Given
two strings a and b, it is dened as the average between the percentage of characters in a
which have (withing a maximum distance of
1
2
max(|a|, |b|)
−1) a matching character
in b, the percentage of characters in b which have a matching character in a, and the
percentage of matching characters which are exactly at the same position in a and b.
The number of matching characters within the distance threshold is denoted by m, the
number of matching characters which are not exactly at the same position is denoted by
48 Chapter 2. Map-Matching Schedule Data
t and often called the number of transpositions. The Jaro Similarity is then dened as
sim
J
(z
a
, z
b
) =
1
3
m
|n
a
|
+
m
|n
b
|
+
m −t
m
. (2.38)
Jaro-Winkler Similarity
To give strings which match from the beginning a slight advantage, Winkler [148] pro-
posed to combine the Jaro Similarity sim
J
(z
a
, z
b
) with 1 − sim
J
(z
a
, z
b
), weighted by the
length l < 5 of the common prex (we check only up to character 4 if the prexes match)
and a scale factor p <=
1
4
. It is then called the Jaro-Winkler Similarity.
sim
JW
(z
a
, z
b
) = sim
J
(z
a
, z
b
) + lp
1 − sim
J
(
z
a
, z
b
)
. (2.39)
Note that p cannot exceed
1
4
, because otherwise the similarity may be greater than 1. In
practice, p is usually set to 0.1.
Jaccard Index
Another class of string similarity measures is based on tokens. While the PED and/or
the SED similarity can account for missing tokens at the beginning or the end of station
labels, it doesn’t handle cases well where tokens are completely scrambled (or inverted).
For example, for n
a
=“Hauptbahnhof Freiburg” and n
b
=“Freiburg Hauptbahnhof”, both
sim
PED
(z
a
, z
b
) and sim
PED
(z
a
, z
b
) are only 0.57, despite the fact that the station labels are
obviously equivalent.
A token-based similarity measure for strings is the Jaccard Index of their (unique)
word sets. Let A be the set of words in station label n
a
, and B be the set of words in
station label n
b
. Then a Jaccard Index based classier is dened as
sim
JAC
(z
a
, z
b
) =
|A ∩ B|
|A ∪ B|
. (2.40)
With the example from above, sim
JAC
(z
a
, z
b
) = 1.
Best Token Subset-Permutation Similarity
With a token-based Jaccard Index, the impact of each missing token is the same, regard-
less of its length. It is also not very robust against minor spelling dierences or missing
location speciers. We propose a similarity score that tries to combine the advantages
of the Jaccard Index (token ordering does not matter) and the edit distance similarity.
Given two labels n
a
and n
b
as well as their unique token sets A and B, this best token
2.8 Advanced Station Similarity Classication 49
subset-permutation similarity (BTS) can be dened as
sim
BTS
(z
a
, z
b
) = max
max
a∈P(A)
sim
∗
ED
(a, n
b
), max
b∈P(B)
sim
∗
ED
(b, n
a
)
, (2.41)
where sim
∗
ED
is the edit distance similarity directly on strings and P(S) is the set of all
possible unique permutations of S with size 1 ≤ n ≤ |S |, concatenated with a space (in
other words, all space-separated concatenated possible permutations for each U ∈ P(S)
with a size > 1, where P(S) is the power set of S). For example,
P
({
“Freiburg”, “Hauptbahnhof”
})
={“Freiburg”, “Hauptbahnhof”,
“Freiburg Hauptbahnhof”,
“Hauptbahnhof Freiburg”}.
The BTS is then the best edit distance similarity between any such permutation of A and
n
1
, or of any such permutation of B and n
a
. An obvious drawback of this measure is its
high calculation cost for labels with many tokens, as |P(S)| grows super-exponentially.
TFIDF
A standard method in Information Retrieval to compute relevance scores for terms in
documents are TFIDF scores [100], a combination of the term frequency (the number of
times a term t occurs in a document d) TF and the document frequency (the number of
documents d ∈ D a term t occurs in) DF. In our case, a term t is a single word (token) in
a station label, a document d is a single station identier label, and the training data (a
list of station labels) constitutes the document collection D 3 d.
Using the document frequency, we can derive a measure of how rare a term is by
taking the quotient of the number of documents |D| and the document frequency. This
quotient will be high if the term occurs only in very few documents, and 1 if it occurs
in every document. To avoid extremely high values of this inverse document frequency
(IDF), it is logarithmically scaled:
idf(t, D) = log
|D|
|{
d ∈ D : t ∈ d
}|
. (2.42)
The TFIDF score of a term, a document and a document collection is then the product
of the TF and the IDF. Given a collection of station labels, we can thus derive relevance
vectors v for each label. A vector v contains TFIDF scores for each token in the label.
As a similarity measure sim
TFIDF
(z
a
, z
b
) we then take the cosine similarity between the
50 Chapter 2. Map-Matching Schedule Data
relevance vectors a and b for n
a
and n
b
:
sim
TFIDF
(z
a
, z
b
) =
a · b
kakkbk
. (2.43)
We note that it may happen that a station label contains tokens that were not encoun-
tered in the training data. Then the DF is 0, and the IDF is not well-dened. For such
cases we simply set the DF to 1, which means an IDF of log |D|.
Combined Classiers
To be able to combine the similarity measures described above, we use the following
simple approach to map them onto a “non-similar” interval of [0, 0.5[ and a “similar”
interval of [0.5, 1]:
sim
0
(z
a
, z
b
) =
1
2
+
sim(z
a
,z
b
)−t
2(1−t)
if sim(z
a
, z
b
) > t,
sim(z
a
,z
b
)
2t
else.
(2.44)
Dierent similarity measures may then for example be combined using soft voting (av-
erage) or hard voting (majority vote).
Machine Learning
Determining the optimal threshold values for the similarity measures described above
can already be considered a simple form of machine learning with a linear classier.
Using TFIDF scores, we additionally “learn” the signicance of tokens, but only on a
global level: a token that may be highly signicant regionally may appear very often
globally, or vice versa. Our evaluation in Section 2.10.1 will give false negative and false
positive examples for a TFIDF based classier where this causes problems. It would also
be preferable if our classier could learn common abbreviations or common spelling
mistakes. Also, the presence of certain tokens (for example “Main Station”) indicates a
major station for which the distance between similar station identiers may be signi-
cantly larger than usual. In this section we develop a classier which is able to consider
these aspects. We extract the following features from our training data of similar and
non-similar pairs {s
a
, s
b
} of station identiers:
1. The great-circle distance between the station identiers in meters.
2. Let n
t
(z) be the number of occurrences of trigram t in the label of station identier
z (for example, n
rei
(z) = 2 if the label of z is “Freiburg im Breisgau”). For the
top-k trigrams in the training dataset, we then add individual features and take
n
t
(z
a
) − n
t
(z
b
) as a feature value. We pad the tokens with spaces on both sides
to encode the beginning and ending of tokens and to ensure that single character
tokens are represented by a single trigram.
2.9 Map-Matching Speedup Techniques 51
Table 2.2: Example feature vectors for three station pairs: (1) “Freiburg im Breis-
gau Hauptbahnhof” @ (47.9966, 7.8404) vs. “Hauptbahnhof” @ (47.9965, 7.8407).
(2) “Okenstraße” @ (48.0105, 7.8545) vs. “Nordstraße” @ (48.0111, 7.8541). (3)
“ZOB” @ (47.9959, 7.8405) vs. “Zentraler Omnibusbahnhof, Freiburg im Breisgau” @
(47.9960, 7.8407).
d
m
d
3д
x
0
y
0
x
1
y
1
rei tra raß aße urg bur ibu ␣Fr Fre eib rg␣ eis Bre sga isg simi
24 20 133 196 133 195 -2 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 yes
72 10 133 196 133 195 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 no
12 47 133 196 133 195 2 1 0 0 1 1 2 1 1 1 1 1 1 1 1 yes
3. Let T (z) be the set of all trigrams of the again space-padded tokens of station iden-
tier z. To capture dierences in trigrams which are not in the top-k, we use the
number of trigrams which are present in one station identier, but missing in the
other as a feature. This is |T (z
a
) ∪T (z
b
)| − |T (z
a
) ∩T (z
b
)|.
4. We take the centroid c of z
a
and z
b
as a representation for the general region the
stations are in. We then take n grids G
0
, . . . , G
n
, all with the same grid cell width w
and height h. The origin for a grid G
i
is osetted by i(w/n) in the x direction, and
by i(h/n) in the y direction. The grid cell coordinates of c on each grid G
i
are then
added as a feature pair (x
i
, y
i
).
The last feature requires some additional clarication. Consider the example given
in Figure 2.19. The intuition behind these interwoven grids is to give the classier
the ability to extract dierent levels of locality through the interference of selected
grids. For example, a combination of grid coordinates (for example д
0
= (133, 197), д
1
=
(133, 196), д
2
= (132, 196)) may encode a region much smaller than an entire grid cell.
Another goal was to soften the eect of the hard grid boundaries.
Table 2.2 gives three example feature vectors. For our evaluation, we trained a ran-
dom forest classier on these features.
2.9 Map-Matching Speedup Techniques
We recall from Section 2.5.4 that we determine the optimal hidden state sequence via
a shortest path search in a layered graph weighted with log probabilities, which can be
done in O(|V |+|E|)as this graph is a DAG. The bottleneck of our map-matching approach
is then the calculation of the |H
i
|·|H
i+1
| shortest paths between hidden state layers, which
are required to calculate the transition probabilities. As schedule data typically only has
to be map-matched in bulk and is updated only a few times a year, the running time
of our approach may seem insignicant at rst. However, real-word schedule data may
consist of millions of vehicle trips. Eciency thus has to be considered.
52 Chapter 2. Map-Matching Schedule Data
133, 196
134, 196
133, 197
134, 197
133, 196
134, 196
133, 197
134, 197
133, 196
134, 196
133, 197
134, 197
𝑎
𝑏
𝑐
𝐺
0
𝐺
1
𝐺
2
Figure 2.19: Three interwoven grids G
0
,
G
1
and G
2
used to encode the approxi-
mate region of coordinates on three lev-
els of locality. Rectangle a, specied by
grid coordinates д
0
= (133, 197), д
1
=
(133, 196), д
2
= (132, 196)), describes a re-
gion on the most local level 0, rectangle b
(д
0
= (134, 196), д
1
= (134, 196)) a region
on level 1, and individual grid cells de-
scribe a region on level 2. There is no com-
bination of grid coordinates to describe c.
2.9.1 Faster Layer-to-Layer Shortest Path Calculations
Calculating all shortest paths in the underlying transportation network between sets
H
i
and H
i+1
is an instance of a many-to-many shortest path search as described in Sec-
tion 1.3.3. As mentioned there, a straightforward way to speed up this calculation is
to not calculate |H
i
| · |H
i+1
| node-to-node shortest paths using Dijkstra’s algorithm, but
only |H
i
| node-to-many shortest paths, namely for each h
i
∈ H
i
all shortest paths from
h
i
to H
i+1
. This can be achieved in |H
i
| Dijkstra runs.
Speeding up the calculation of many-to-many shortest paths is an active research
topic. However, we show in this section that (somewhat surprisingly), a single Dijkstra
run is enough to calculate all the relevant shortest path costs between H
i
and H
i+1
in our
HMM based on shortest travel times. That is, we prove that (1) for some edges from H
i
to H
i+1
in the layered graph we do not require the exact edge weights to nd the most
likely hidden state sequence, and (2) a single (slightly modied) set-to-many shortest
path search between H
i
and H
i+1
using Dijkstra’s algorithm is enough to calculate the
required weights. As mentioned above, this extends a similar technique from [52].
For simplicity, we rst assume that the shortest path costs in the underlying trans-
portation network are equivalent to the transition edge costs in the layered graph.
Lemma 2.6. Let H
i
be the set of all nodes corresponding to hidden states of layer i. By c
i
j
we denote the cost of the shortest path from the start node q
0
to node h
i
j
∈ H
i
. Assume that
all c
i
for H
i
are known. Then for each h
i+1
k
∈ H
i+1
, c
i+1
k
= min{c
i
j
+ w(h
i
j
, h
i+1
k
) | h
i
j
∈ H
i
},
where w(h
i
j
, h
i+1
k
) is the weight of edge (h
i
j
, h
i+1
k
). In other words, the cost c
i+1
k
of the shortest
path from q
0
to h
i+1
k
will be the smallest sum of an edge weight back to layer i and the
known shortest path cost from q
0
to corresponding node in layer i.
Proof. The proof is straightforward: per the construction of c
i+1
k
, there cannot be any
cheaper path from q
0
.
2.9 Map-Matching Speedup Techniques 53
H
1
H
3
h
1
2
h
1
1
h
2
1
h
2
2
h
2
3
q
0
t
h
3
2
h
3
3
h
3
1
0
0
0
0
0
1
(6)
(4)
5
3
(9)
N
v
∗
1
3
5
H
2
Figure 2.20: Calculation of the required transition shortest path costs between two
layers H
2
and H
3
through the transportation network N, under the assumption that we
ignore emission probabilities. We add a virtual node v∗connected via virtual edges to all
h
2
∈ H
2
and initialize their edge weights to the shortest path cost from q
0
to respective
h
2
. The edge weights of the previous transitions from layer H
1
to layer H
2
that were not
required to calculate the shortest path to the respective h
2
∈ H
2
are given in brackets,
and the required edge weights for the transitions from H
2
to H
3
can be found by a single
Dijkstra run on the transportation network N from v
∗
to H
3
.
This means that for some node h
i+1
k
∈ H
i+1
, we only require the cost of the shortest
transitional path starting at some node c
i
j
in H
i
(osetted by its known shortest path cost
c
i
j
) in the underlying transportation network to determine the shortest path cost c
i+1
k
in
our HMM layer graph.
Lemma 2.7. We can calculate all shortest paths p
k
from H
i
to h
i+1
k
(each p
k
minimizes
c(p
k
) + c
i
j
) with a single run of Dijkstra’s algorithm if we initialize the priority queue with
each h
i
j
∈ H
i
and its known shortest path cost c
i
j
as a cost label.
Proof. The proof is similar to the correctness proof of Lemma 1.3. We add a new virtual
node v
∗
, connect it to each h
i
j
∈ H
i
with a virtual edge e
j
and set the corresponding
edge weight to c
i
j
. A single run of Dijkstra’s algorithm from v
∗
then yields for each node
h
i+1
k
∈ H
i+1
a path p
k
which consists of a virtual edge e
j
with weight c
i
j
and the shortest
path from h
i+1
k
(if it would not be the shortest path, then p would not be a shortest path).
This path p
k
thus minimizes c(p
k
) + c
i
j
. Adding each h
i
j
∈ H
i
to the priority queue with
cost label c
i
j
implicitly constructs both v
∗
and the virtual edges.
Figure 2.20 gives an example.
54 Chapter 2. Map-Matching Schedule Data
So far, our assumption was that the edge weights in the layered graph are equivalent
to the shortest path cost of the corresponding transition. This is not true in general.
First, we have to consider the costs associated with the emission probabilities for the
h
i+1
∈ H
i+1
. Luckily, we can simply add the emission weight associated with a hidden
state h
i
to its oset cost c
i
j
. Note that this means that we have to manually add the
emission weights for the hidden states of the last layer later on.
Second, we note that the transition probability does not directly depend on the short-
est path costs in the underlying transportation network, but is a function of it. Let now
c
i
j
again be the shortest path cost through the layered graph to hidden state node h
i
j
, let
e
i
j
be the emission probability of h
i
j
(in log space), let d be the shortest path cost from h
i
j
to h
i+1
k
, and let w(d) be the transition probability from h
i
j
to h
i+1
k
based on the shortest
path cost (also in log space). The Dijkstra run above then yields for each hidden state
h
i+1
k
the cost c
i+1
k
= min{c
i
j
+ e
i
j
+ d | h
i
j
∈ H
i
} (note that we can easily extract the value of
d from the calculated shortest path, as c
i
j
+ e
i
j
is completely contained in the virtual edge).
To be sure that this indeed corresponds to the optimal path if we would not consider d,
but w(d), w must be monotonic, that is for two shortest path lengths d
1
and d
2
, it must
hold that
d
1
< d
2
⇔ w(d
1
) < w(d
2
). (2.45)
For the transition probabilities described in Section 2.5.2 which were modeled as an
exponential distribution, this is obviously the case. Using this model, we can there-
fore calculate the most likely hidden state sequence for m observations in O(m(|E
N
| +
|V
N
|log |V
N
|)), where E
N
and V
N
are the edges and nodes of the underlying transporta-
tion network N. The naive method of calculating H
i
×H
i+1
node-to-node shortest paths
would have required an asymptotic running time of O(mn
2
(|E
N
| + |V
N
|log |V
N
|)). The
straightforward optimization of only calculating n node-to-many shortest path has an
asymptotic running time of O(mn(|E
N
| + |V
N
|log |V
N
|)). We will evaluate the eects of
this technique in Section 2.10.3.
2.9.2 A
∗
Heuristic
A
∗
heuristics as described in Section 1.3.3 are typically given for single target nodes,
but it is easy to see that they can also be used in multi-target scenarios (one-to-set, set-
to-set, one-to-many, or set-to-many). Their eciency, however, greatly depends on the
characteristics of the target set.
Lemma 2.8. LetT be a set of no des. We add a virtual node ω and edges (t, ω)with w(t, ω) =
0 for each t ∈ T . Let h then be a heuristic function for target node ω. If h is an admissible
heuristic for ω, it is an admissible heuristic function for each t ∈ T . If h is a consistent
heuristic for ω, it is a consistent heuristic for each t ∈ T .
2.9 Map-Matching Speedup Techniques 55
Proof. For h to be admissible for each t ∈ T , the following must hold: ∀u ∈ V , ∀t ∈ T :
h(u) ≤ c(u, t), where c(u, t) is the shortest path cost from u to t. As h is admissible for ω,
it holds that ∀u ∈ V : h(u) ≤ c(u, ω). But as w(t, ω) = 0∀t ∈ T , c(u, ω) = c(u, t)∀t ∈ T , it
also holds for each t ∈ T . For h to be consistent for each t ∈ T , the following must hold
for each pair of nodes v, u: ∀t ∈ T : h(v) ≤ w((v, u)) + h(u). But this is already the case
as h was assumed to be consistent.
Assume now that we are searching for the shortest path(s) from some node s into a
set of target nodesT in a network graph N for some vehicle type. Let C be the connected
component of N in which s lies and note that any target node t must also be contained
in C, or otherwise the shortest path does not exist.
Let now v
max
be the maximum possible speed in C, let z be a virtual node positioned
at the centroid of all t, and let R = max{d(t, z) | t ∈ T } be the maximum meter distance
between z and any t (d(u, v) gives the meter distance between two nodes u and v). We
then use the following simple heuristic function:
h(v) = max
0,
d(v, z) − R
v
max
. (2.46)
Lemma 2.9. h(v) is admissible for all t ∈T , that is ∀v ∈ V : h(v) ≤ c(v, s)
Proof. Recall that we model our edge weight function on the travel time and that the
punishment for no matching line as described in Section 2.7.1 increases travel times. Let
c(u, v) be the shortest path cost from u to v. Then
c(u, t) ≥
d(v, t)
v
max
(2.47)
≥
d(v, z) − R
v
max
(2.48)
≥ max
0,
d(v, z) − R
v
max
= h(v). (2.49)
Lemma 2.10. h(v) is consistent, that is h(v) ≤ w((v, u)) + h(u) for adjacent nodes v, u.
Proof. We observe that
d(v,u)
v
max
≤ w((v, u)) and prove the stronger condition
h(v) ≤
d(v, u)
v
max
+ h(u) (2.50)
⇔ max
0,
d(v, z) − R
v
max
≤
d(v, u)
v
max
+ max
0,
d(u, z) − R
v
max
(2.51)
⇔ max
{
0, d(v, z) − R
}
≤ d(v, u) + max
{
0, d(u, z) − R
}
. (2.52)
56 Chapter 2. Map-Matching Schedule Data
There are now 4 cases: if only the left-hand d(v, z) − R falls below zero, we get 0 ≤
d(v, u) + d(u, z) − R, which is obviously always true as we assumed d(u, z) − R ≥ 0. If
only the right-hand d(u, z)−R falls below zero, then u is either inside the circle of radius
R around z, or on the border of it. We get d(v, z) − R ≤ d(v, u), which is again true as
d(v, z)−R is exactly the distance from v to this circle. If both the right-hand d(u, z) −R
and the left-hand d(v, z) − R fall below zero, we get 0 ≤ d(v, u), which is also always
true. If neither fall below zero, we have
d(v, z) − R ≤ d(v, u) + d(u, z) − R (2.53)
⇔ d(v, z) ≤ d(v, u) + d(u, z), (2.54)
which is true as the meter distance between nodes (we use the haversine formula to
calculate them) satises the triangle inequality.
As h(v) is hence consistent, the A
∗
algorithm will only expand a node u exactly once,
and as h(v) ≥ 0 and c(u, v) ≥ 0, all nodes relaxed while expanding u will get a new
label that is higher than or equal to the last extracted label of u. The priorities extracted
from our priority queue Q thus form a monotonically increasing sequence, and we may
therefore use a monotone priority queue to get some additional performance gains. In
our reference implementation, we used a priority queue based on a radix heap [3].
The eciency of this heuristic greatly depends on the average distance of visited
nodes to the circle with radius R around z, and thus on the maximum distance for candi-
date nodes around the sample point. If all visited nodes are inside the circle, the heuristic
will always be 0, at which point we will be running a raw Dijkstra (with the additional
overhead of calculating h(v) for each relaxation). This might for example be the case for
very short bus routes, where most of the source hop nodes are already inside this circle.
On the other hand, for longer routes, we expect this blurring eect to be insignicant
(this will be conrmed by our experimental evaluation in Section 2.10.3).
2.9.3 Early Stopping
A straightforward speedup technique is to not further expand nodes labeled with a cost
above some threshold t during the Dijkstra run. For example, if a target node candidate
set H
i
contains some node h which is in the same connected component as the source
node, but is only reachable through a complicated detour, large parts of the graph will
be explored to nd the shortest path to h, only for it to later yield a transition probability
which is orders of magnitudes smaller than other transition probabilities into H .
As we are dealing with schedule data, we already have a rough upper bound on the
time cost of the shortest path - for a stop transition from i − 1 to i in some a trip T ,
this is simply the dierence ∆
i
= t
↓
i
−t
↑
i−1
between the arrival time at the stop i and the
departure time at the last stop i −1. Simply taking this time dierence ∆
i
as the threshold
2.9 Map-Matching Speedup Techniques 57
t would be too aggressive, as schedule times are usually given with a granularity of
one minute, rounded to the nearest minute, and the scheduled travel time between two
nearby stops is often 0 for urban bus or rail routes. Another problem is that we might also
underestimate edge travel times. In our reference implementation, we used t = 60 + 3∆
i
seconds as a threshold.
2.9.4 Bulk Processing
In this section, we describe several speedup techniques that use the fact that schedule
data is usually map-matched in bulks. We will consider the trivial speedup technique
of matching two trips that only dier in their starting time oset once as given and
therefore assume that all trips are unique in this regard.
Shortest-Path Caching
Even if we consider trips to be unique, many scheduled trips will typically share station-
to-station segments. Thus, many of the shortest paths required to determine the tran-
sition probabilities will occur over and over again during the map-matching process.
A straightforward idea is to only compute their cost once, store it, and use the cached
result later on. We use this approach as a baseline speedup technique in our evaluation.
Faster Bulk Processing via Trip Tries
Shortest-path caching is orthogonal to the technique described in Section 2.9.1, where
we explicitly avoided calculating too many transitional shortest paths. If we recall our
approach there, we realize that to compute the required edge weights between H
i
and
H
i+1
, the only used information external to the currently processed layers was the short-
est path costs c
i
j
through the layered graph to nodes h
i
j
∈ H
i
. But if two trips have the
same attributes, and if they serve the same stations up until s
i
, the c
i
j
will be exactly the
same, regardless of the further course of the trip. We may thus re-use them. To e-
ciently calculate, store, and process prex overlaps of trips, we rst compute a forest of
trip prex trees, which we call trip tries, and perform the map-matching on them.
Denition 2.10 (Trip Trie). Let B = (S, L, Λ, T) be a Bag of Trips according to Deni-
tion 1.1, and let T
0
⊆ T be a subset of trips in which for each pair t
1
, t
2
∈ T it holds that
l
1
= l
2
(that is, they all have the same line). A trip trie for T
0
is a prex tree in which
each node v is labeled with a station s ∈ S. Each node v is additionally labeled with trips
T
0
v
⊆ T
0
, and for each t ∈ T
0
v
the following holds: the station labels of the (ordered)
nodes on the path from v to the tree root form a prex of the station sequence Σ
t
.
Figure 2.21 gives an example of a trip trie for a typical (simplied) bus line with 6
line course variants, served by vehicle trips t
1
, t
2
, t
3
, t
4
, t
5
, t
6
. Although trips t
2
, t
3
, t
4
, and
58 Chapter 2. Map-Matching Schedule Data
s
1
s
3
s
2
s
3
s
5
s
6
s
5
s
4
t
1
t
3
t
4
t
5
t
2
l
1
t
6
H
1
H
2
H
3
H
3
H
4
H
5
H
5
H
6
Figure 2.21: Left: A trip trie for a subset T
0
= {t
1
, t
2
, t
3
, t
6
, t
5
, t
6
} ⊆ T in which all trips
have line l
1
. Each node is labeled with a scheduled station and (optionally) a trip ending
at this node. Note that the same station may appear in multiple nodes, as is the case for
s
3
and s
5
here. Right: Schematic view of the layered graph corresponding to our hidden
Markov model, white nodes are the hidden states (the station candidates), and arrows
denote hidden state transitions whose weights depend on the shortest path connecting
the station candidates.
t
5
have dierent stops, we can not only re-use the shortest path distances through the
transportation network N for the transitions from H
1
to H
3
and from H
3
to H
4
, we may
even apply the speedup technique from Section 2.9.1. For real-world schedule data with
many dierent lines, we typically have to build a substantial number of trip tries.
Denition 2.11 (Trip Trie Forest). Given a Bag of Trips B, a trip trie forest contains a
trip tree for each line l ∈ L.
In this denition, the tip trie ignores travel times, but it is easy to see that we can
incorporate them by not only labeling a tree node v with a station s, but with a pair
(s, δ), where δ is the scheduled travel time from the parent node of v to v.
Node-Candidate Predetermination
It is also evident that we do not need to determine the hidden states when we build the
trip trie - given a station s and a transportation network N, the edge candidates and
their emission probabilities will always be the same, regardless of the scheduled stops of
a trip or the line attributes. We can therefore determine all candidates for all stops once,
prior to building the trip trie.
2.10 Experimental Evaluation 59
2.10 Experimental Evaluation
We evaluated both our map-matching approach and the advanced station similarity clas-
sication on several real-world datasets. In Section 2.10.1, we rst present the evaluation
results of the station similarity classication methods discussed in Section 2.8. In Sec-
tion 2.10.2, we then evaluate variants of our map-matching approach against a baseline
and the widely-used approach from [109]. In particular, we also measure the impact of
the station similarity classication on the map-matching quality. In Section 2.10.3, we
then evaluate the overall speed of our approach and the impact of the speedup techniques
described in the previous section.
2.10.1 Quality of the Station Similarity Classication Methods
This section gives an overview over the results of an extensive evaluation of station
similarity classication methods on two ground truth datasets extracted from OSM, with
2.1 million and 13.6 million station identier pairs, respectively. Preliminary results have
been previously published in [15] and formed the basis of a quality assessment tool for
public transportation stations in OSM (staty
6
), which has been described in [16]. For this
work, we re-ran the evaluation on updated OSM data. The ground truth datasets, pre-
trained models and a classication playground are available online
7
, as is the evaluation
setup
8
.
Evaluation Setup
We built two ground truth datasets from OSM data for the following regions: the British
Isles (BI), and Germany, Austria, and Switzerland (DACH). The dataset dimensions are
given in Table 2.3. Our approach was as follows: for each node describing a public trans-
portation stop or station (we lter for all key/value pairs describing station nodes for var-
ious methods of transportation described in the OSM wiki
9
), we extracted labels given in
the OSM attributes name, ref_name, uic_name, official_name, alt_name, loc_name,
reg_name, short_name, and gtfs_name. Each unique label together with the node’s lat-
itude/longitude position was then added as a station identier. Two station identiers z
a
and z
b
that were within 1,000 meters were added as similar to the ground truth data if ei-
ther: (1) z
a
and z
b
came from the same node n. (2) z
a
and z
b
came from dierent nodes n
a
and n
b
, but n
a
and n
b
were part of the same OSM relation public_transport=stop_area
(which groups dierent parts of a larger public transport stop
10
).
6
https://staty.cs.uni-freiburg.de/
7
https://staty.cs.uni-freiburg.de/datasets/
8
https://github.com/ad-freiburg/statsimi-eval
9
https://wiki.openstreetmap.org/wiki/Public_transport
10
https://wiki.openstreetmap.org/wiki/Tag:public_transport%3Dstop_area
60 Chapter 2. Map-Matching Schedule Data
Table 2.3: Dimensions of our ground-truth datasets for the British Isles (BI) and Ger-
many, Austria, and Switzerland (DACH). N is the number of stations in the OSM data, G
the number of station group relations, N
0
is the number of stations without any group
relation, |s | the number of unique station identiers, д is the average group relation size,
d
+
is the average distance (in meters) between positive ground truth pairs, K
−
is the
number of extracted pairs that were not similar, and K
+
is the number of similar pairs.
The numbers in this table are without the spicing described in Section 2.10.1.
N G N
0
|s| д d
+
K
−
K
+
K
BI 270k 15k 234k 261k 3.7 56.7 1.7M 0.4M 2.1M
DACH 679k 102k 350k 875k 5 46.1 11.1M 2.6M 13.6M
Two station identiers z
a
and z
b
within 1,000 meters were added as not similar to
the ground truth data if z
a
and z
b
came from dierent nodes n
a
and n
b
, but n
a
and n
b
were members of dierent OSM relations public_transport=stop_area. If in such a
situation one or both of n
a
and n
b
were not part of any public_transport=stop_area
relation, we did not add {z
a
, z
b
} as a non-similar pair. The reason is that forgetting to
group similar station nodes by a public_transport=stop_area relation is common in
OSM. We ignored station identier pairs that were more than 1,000 meters away, as we
could not nd any similar station identier pair with a distance greater than 1,000 meters
in our OSM datasets, except for some obvious mapping mistakes.
For all such ground truth pairs, we additionally performed the following sanity checks:
if a pair {z
a
, z
b
} was considered not similar, but z
a
and z
b
had exactly the same la-
bel and were within 250 meters, we ignored this pair. If z
a
and z
b
were in dierent
public_transport=stop_area relations r
a
and r
b
, but r
a
and r
b
were themselves in a
relation public_transport=stop_area_group (seldomly used to add an additional hi-
erarchical level for very large stations), we also ignored this pair.
As OSM data contains only very few obvious mistakes, we spiced our ground truth
dataset by randomly adding obvious “not similar” pairs. With probability p = 0.5,
we selected for a station identier z
a
from the OSM data 5 random station identiers
{z
1
b
, z
2
b
, z
3
b
, z
4
b
, z
5
b
} which were more than 1,000 meters away from s (so that we could be
sure that they were not similar). For each such z
b
, we added a new station identier z
0
b
which had the same label as z
b
, but was given a new random coordinate within 100 me-
ters of s
a
. The pair {z
a
, z
0
b
} was then added as “not similar”. To also simulate coordinate
imprecision, we randomly selected similar station pairs and added some Gaussian noise
to the coordinates of one station (we used a standard deviation of 100 meters).
The ground truth dataset was then divided into a random selection of 20% of the
data as a training set, and a random selection of 80% of the data as a testing set. The
immense size of the ground truth datasets allowed for such a small training set, and
preliminary experiments showed only minimal gains when larger training sets were
2.10 Experimental Evaluation 61
Table 2.4: Station similarity classication evaluation results.
BI
method t prec. rec. F1
P 100 m 0.66 0.93 0.77
ED 0.85 0.99 0.85 0.91
PED 0.85 0.93 0.88 0.91
J 0.9 0.98 0.85 0.91
JW 0.9 0.93 0.88 0.91
JAC 0.65 0.96 0.86 0.91
BTS 0.85 0.91 0.9 0.91
TFIDF 0.99 0.99 0.83 0.9
P+ED 40 m + 0.6 0.96 0.9 0.93
P+JAC 80 m + 0.55 0.95 0.93 0.94
P+BTS 10 m + 0.5 0.93 0.9 0.91
P+TFIDF 40 m + 0.2 0.95 0.93 0.94
RF — > 0.99 0.98 > 0.99
DACH
t prec. rec. F1
125 m 0.4 0.97 0.57
0.85 0.99 0.66 0.79
0.9 0.92 0.73 0.81
0.85 0.93 0.7 0.8
0.9 0.9 0.71 0.8
0.45 0.84 0.88 0.86
0.85 0.92 0.92 0.92
0.7 0.9 0.84 0.87
40 m + 0.55 0.9 0.82 0.86
90 m + 0.45 0.92 0.93 0.92
20 m + 0.65 0.92 0.91 0.92
60 m + 0.5 0.94 0.93 0.94
— > 0.99 > 0.99 > 0.99
chosen. All classiers were then trained on the training datasets, and evaluated against
the test datasets (for the similarity measure based classiers, training meant nding the
optimal threshold values).
Our evaluation metrics were based on the number of true positives (TP), the number
of true negatives (TN), the number of false positives (FP), and the number of false neg-
atives (FN). Namely, we used precision (
TP
TP+FP
), recall (
TP
TP+FN
), and F1 scores, where the
latter is the harmonic mean between precision and recall.
Results
Our results, averaged from 5 runs, are reported in Table 2.4. We evaluated each stan-
dalone classier on both test datasets. Using the technique described in Section 2.8.2,
we also evaluated the following combinations of classiers: P+ED, P+JAC, P+BTS, and
P+TFIDF, and a random forest (RF) based classier using the features described in Sec-
tion 2.8.2 (we used the random forest classier provided by the Python scikit-learn
library with default parameters). For the feature vectors, we used the top 2,500 trigrams
and 2 grids G
0
and G
1
, with cell widths and heights chosen such that the earth was
covered by 256 × 256 cells.
For both datasets, the random forest based classier (RF) clearly outperformed each
other classier, achieving F1 scores > 0.99. For the combined classiers, two winners
emerged: P+TFIDF and P+JAC, but only by a small margin to P+ED and P+BTS. In partic-
ular, a simple combination of the geographical distance and the meter distance (P+ED)
62 Chapter 2. Map-Matching Schedule Data
FN
“Parkweg” @ (52.0149, 7.2051)
“Rosendahl, Osterwick, Parkweg” @ (52.0149, 7.2051)
FP
“Bruck an der Mur” @ (47.4136, 15.2793)
“Bruck an der Mur, Waldweg” @ (47.4185, 15.2736)
Figure 2.22: A typical false negative and a typical false positive given by a Jaccard
index based classier. The location speciers “Rosendahl, Osterwick” prevents a positive
classication in the rst example. In the second example, a train station labeled after a
town was positively matched against a bus stop “Waldweg” provided with the town
name as a location specier.
FN
“Bromley-By-Bow Platform 2” @ (51. 5248, −0.0115)
“Bromley By Bow Station” @ (51.5234, −0.0121)
FP
“Clapton Girls’ Academy” @ (1.5539, −0.0537)
“Clapton” @ (51.5617, −0.0568)
Figure 2.23: A typical false negative and a typical false positive given by a PED based
classier.
already achieved an F1 score of 0.93 on the BI dataset, but only of 0.86 on the DACH
dataset. Upon manual investigation, we found that station identier names were much
more uniform and standardized on the British Isles than they were in German-speaking
countries. This was also reected in the high performance of the ED standalone classier
on the BI dataset, which already achieved an F1 score of 0.91, but only 0.79 on the DACH
dataset. Because of this station name uniformity, all similarity measures based on string
similarity performed equally well on the BI dataset, achieving F1 scores between 0.9 and
0.91. For these measures, the dierences on the DACH dataset were much greater: F1
scores ranged from 0.79 to 0.92, with the BTS classier achieving the best result.
The reasons for false positives and false negatives for the standard string similarity
based classier were as expected. Figure 2.23 gives an example of a typical FN and FP
for the PED classier and Figure 2.22 shows typical such examples for the JAC classier.
For the TFIDF based classier, the reasons for erroneous classication were more inter-
esting and subtle. Consider for example Figure 2.24. Here, one problem was that the
TFIDF classier learned that some tokens have very little relevance on a global (or, in
this case, national) level, but these tokens had high relevance on a regional or local level.
These tokens were typically locations which are present in most villages, like cemeter-
ies, schools, churches, or town halls. On the other hand, the classier had a tendency to
give village and town names which were unique nationally too much signicance.
Finally, we found the following main causes for the remaining false negatives and
false positives of the random forest classier: (1) extreme outliers, (2) stations at street
2.10 Experimental Evaluation 63
FN
“Auerbach (Karlsbad), Rosenweg” @ (48.9161, 8.5341)
“Rosenweg” @ (48.9160, 8.5343)
FP
“Cobus, Kiekebusch Alte Schule” @ (51.7215, 14.3646)
“Kiekebusch Friedhof, Cobus” @ (51.7179, 14.3672)
Figure 2.24: A typical false negative and a typical false positive given by a TFIDF based
classier. Because the input DACH dataset was large, the TFIDF scores gave the nation-
ally common token “Schule” (school) and “Friedhof” (cemetery) only little signicance,
although they are highly signicant locally. On the other hand, the TFIDF scores gave
the village name “Auerbach” which is rare nationally, too much signicance, although
many stations in this region are prexed by it.
FN
“Lile Ilford School” @ (51.5483, 0.0577)
“Church Road” @ (51.5479, 0.0569)
FP
“Galsworthy Road/Moonshine Lane” @ (53.4178, −1.4808)
“Moonshine Lane - Galsworthy Road” @ (53.4178, −1.4803)
Figure 2.25: A typical false negative and a typical false positive for our RF based clas-
sier. For the false negative, our model actually found a mapping mistake in OSM. The
false positive are two stations at the same intersection, but in dierent streets. As the
RF classier does not consider the ordering of trigrams, they were classied as similar.
intersections as shown in Figure 2.25, (3) ambiguous cases, and (4) errors in the ground
truth data. As mentioned above, the last point motivated the development of the OSM
quality assessment tool staty
11
, which provides suggestions for grouping and renaming
stations based on the classication results of our RF classier [16].
Choosing the Best Classier for Map-Matching For our map-matching task, there
are three criteria a classier should meet: (1) it must be fast, (2) it must be accurate, and
(3) it should require minimal precomputation, as we want to map-match schedule data
without rst training a classier or learning TFIDF scores from some external dataset.
Based on these criteria, the P+JAC classier was the clear winner. Threshold values
were similar on both datasets, it does not require any precomputation, and it can be
eciently computed (the other contender requiring no precomputation, P+BTS, turned
out to be too costly to compute during the map-matching process). We consequently
used a P+JAC classier with the averaged thresholds from both test datasets, that is 85
meters and a Jaccard index of 0.5. The impact of this station similarity classication on
the map-matching process will be evaluated in the next section.
11
https://staty.cs.uni-freiburg.de/
64 Chapter 2. Map-Matching Schedule Data
2.10.2 Quality of Our Map-Matching Approach
In this section, we evaluate the quality of several variants of our map-matching ap-
proach: a raw version without considering line or station metadata (OURS), a version
with an additional penalty for station candidates classied as “not similar” (OURS+SM),
a version with an additional time penalty for unmatched transit lines in the input data
(OURS+LM), and a version with both the unmatched station candidate and the un-
matched transit line penalty (OURS+SM+LM). We compare them against the best base-
line from Section 2.4, namely the global set-to-set routing approach based on the ap-
proach from [52] (G-STS), and a state-of-the-art map-matching technique from [109]
(DIST-DIFF, also based on a HMM). DIST-DIFF is for example used in the GraphHopper
12
and the OSRM routing engines
13
. DIST-DIFF does not use stations as hidden states, but
road segments, and uses a normal distribution for the emission probabilities. As men-
tioned above, the transition probabilities are based on the dierence between the sample
point distance, and the meter length of the shortest path.
We used the following public transit schedules for our evaluation: the bus and tram
network of Vitoria-Gasteiz, the bus, tram, rail, ferry, and funicular network of Zurich,
the bus, tram, rail, and ferry network of Seattle, the bus, tram, and subway network
of Vienna, the bus, tram, rail, and subway network of Paris, the entire public transit
network of Switzerland, and the entire public transit network of Germany. The datasets
were given in the General Transit Feed Specication (GTFS) format [76]. Table 2.5 gives
an overview over their dimensions.
Evaluation Setup
For the quality evaluation of our map-matching approaches, we removed existing ground
truth shape information from the testing dataset and map-matched this stripped version.
The results were then compared to the original ground truth paths using the metrics
described below.
Finding evaluation datasets which gave a good ground truth to measure the quality
proved to be dicult for several reasons. First, GTFS data is often published with incom-
plete shape information, or none at all (as our survey in Section 2.1.2 clearly showed).
Second, many GTFS datasets lack the optional shape_dist_traveled eld, which gives
each vehicle stop a distinct progression on the shape. Without this eld, it is not pos-
sible to reliably nd out the stop position on the shape (because the stop’s coordinates
are typically placed at the station centroid, snapping the stop to the shape is prone to
errors). Even if shape data with the shape_dist_traveled eld is available, the quality
is often unsatisfactory, either because the sampling rate is very low, or because shapes
are simply incorrect, which would render a quality evaluation meaningless.
12
https://github.com/graphhopper/graphhopper
13
https://github.com/Project-OSRM
2.10 Experimental Evaluation 65
Table 2.5: Dimensions of our testing datasets. Under stations we give the total number
of stations contained in the GTFS feed. Under trips we give the total number of contained
trips. Under unique trips we give the number of trips that only dier in their attributes
(MOT, line name) and their station course, but not in the time oset at the rst station
or the service date. Under tries we give the number of trip tries (unique trips sharing
common station course prexes). Under shapes we denote whether ground truth shape
data was available. The total number of edges over all transportation network graphs
for all MOTs is given under |E|.
stations trips unique trips tries shapes |E|
Vit.-Gast. (VG) 338 4.0k 57 37 • 13.4k
Seattle (SE) 6.7k 27.7k 545 408 • 281.6k
Zurich (ZH) 5.6k 96.1k 1.9k 1.1k • 137.7k
Vienna (V) 4.5k 176.7k 1.1k 664 • 83.9k
Paris (P) 53.7k 307.1k 10.4k 6.3k 830.5k
Switzerland (CH) 39.0k 545.6k 69.8k 42.2k 4.8M
Germany (DE) 501.4k 1.7M 243.5k 109.8k 64.7M
We could not nd a large national schedule dataset matching these criteria. We
therefore evaluated the quality of our approach on a subset of our evaluation datasets
for which good ground-truth data was available: Vitoria-Gasteiz, Seattle, Zurich, and
Vienna, the union of which covers bus, streetcar, subway, rail, funicular, and ferry ser-
vices. For the Switzerland and Germany dataset (both containing the entire nation-wide
schedule data for all methods of transportation), and the Paris dataset, we only evaluated
the speed of our approach (the quality can for example be inspected manually with our
tool TRAVIC
14
). During our experiments, we also noticed that schedule data tting the
criteria described above typically had station coordinates of very high precision. From
our experience, this is an exception in real-world schedule datasets (for example, the
dataset for Germany only has station centroids as coordinates). To measure the qual-
ity of our approaches under imprecise station coordinates as they are typically found
in real-world schedule data, we hence also evaluated all of them under Gaussian noise
added to the station coordinates.
For the Seattle dataset, we found the coverage of public transit station in OSM to be
highly incomplete. We therefore rst snapped the stations from the GTFS dataset onto
the transportation network. The corresponding station node candidates were slightly
penalized during the map-matching process.
To make comparisons fair, we used the same underlying routing engine (imple-
mented by us) for all tested methods. For all methods, we only considered station node
14
https://travic.app
66 Chapter 2. Map-Matching Schedule Data
candidates (G-STS, OURS+*) or road segment candidates (DIST-DIFF) that were within a
distance of 100 meters to the sample points. For the parameter λ
t
of the transition proba-
bilities, we used xed values: For DIST-DIFF, we xed λ
t
using an estimator suggested in
[109]. The estimated parameter corresponded to a mean meter distance of 17.04 meters.
For OURS, OURS+SM, OURS+LM, and OURS+SM+LM, we used a λ
t
that corresponded
to a median travel time between stations of 2 minutes. For the emission probabilities
for all HMM based methods, we used the parameters that optimized the average results
over all test datasets, under no noise. This will be further explained below. The full
evaluation setup can be found online
15
.
Evaluation Metrics
We used two metrics to compare a shape found by our approach to the corresponding
ground truth: the average Fréchet Distance δ
aF
and the percentage A
N
of ground-truth
segments that could not be matched (within a small tolerance) to the corresponding
segments in the map-matched shape.
Average Fréchet Distance The Fréchet Distance δ
F
[63] between two curves P and
Q is a well-established method of measuring their similarity. It is often described as the
minimum length of a leash that allows a person and their dog to do a complete walk on
P and Q. They can both go forward, stop, but may never backtrack.
In its discrete version δ
dF
it denes a coupling C = (p
a
0
, q
b
0
), (p
a
1
, q
b
1
), ..., (p
a
n
, q
b
n
) of
points on p ∈ P, q ∈ Q that minimizes δ
dF
= kC k = max
i
kp
a
i
−q
b
i
k, where kp
a
i
−q
b
i
k
is the euclidean distance. a and b are both monotonously increasing with a step size of
at most 1. To approximate δ
F
, the sampling rate of both P and Q has to be increased. If
the sampling rate goes towards innity, δ
dF
converges to δ
F
[53].
Both δ
F
and δ
dF
give short outliers signicant impact. A metric that better matches
our intuitive understanding of curve similarity and which is used as our rst evalua-
tion metric is the average Fréchet Distance δ
aF
[32]. δ
aF
may be dened as the summed
Fréchet Distance δ
sF
(P, Q)divided by the total length kP k of P: δ
aF
(P, Q) = δ
sF
(P, Q)/kP k.
The summed Fréchet Distance δ
sF
can be approximated as follows [32]:
δ
sF
≈ min
C
(p
a
i
,q
b
i
)∈C
kp
a
i
−p
a
i
−1
k · kp
a
i
−q
b
i
k, (2.55)
which is the minimized sum of the distances between individual couples (p
a
i
, q
b
i
) ∈ C,
weighted by the length of the segment of P dened by p
a
i
and its predecessor p
a
i
−1
.
15
https://github.com/ad-freiburg/pfaedle-eval
2.10 Experimental Evaluation 67
10
30
50
70
90
110
130
2
8
32
128
512
2048
8192
0
0.2
0.4
0.6
0.8
1
σ
noise
σ
e
A
10
N
10
30
50
70
90
110
130
2
8
32
128
512
2048
8192
0
0.2
0.4
0.6
0.8
1
σ
noise
1
λ
e
A
10
N
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Figure 2.26: Left: A
10
N
scores for DIST-DIFF under increasing Gaussian noise (with stan-
dard deviation σ
noise
) for dierent emission probability parameters σ
e
. Right: A
10
N
scores
for OURS under increasing Gaussian noise for dierent emission weights λ
e
.
Non-Matched Hops Our second metric is directly based on the discrete Fréchet dis-
tance δ
dF
. For some segment P
[a,b]
between two stations a and b on the map-matched
path P and the corresponding segment Q
[a,b]
in the ground truth Q, we say that P
[a,b]
matches Q
[a,b]
if δ
dF
(P
[a,b]
, Q
[a,b]
) ≤ 100 meters (rail platforms and bus stops typically
follow the road or tracks for a certain length, with no well-dened stopping position of
vehicles, and the threshold value of 100 meters aims to account for that). With the total
number of hops N and the number of unmatched hops A, we then dene A
N
= A/N .
For example, A
N
= 0.1 means that 10% of the hops in Q could not be matched to the
corresponding hop in P. For our evaluation, we counted the percentage of trips with an
A
N
score below 0.1. We denote this accuracy measure by A
10
N
.
Evaluation Results
Figure 2.26 gives the A
10
N
scores for DIST-DIFF and OURS for various levels of Gaussian
noise, under dierent parameters for the emission probabilities. As described above, we
used xed parameters for the transition probabilities. The scores are averaged over all
test datasets. The evaluation was run 10 times, and the nal values again averaged.
For further evaluation, we used the emission probability parameters that optimized
the average A
10
N
scores under no noise. These values were a standard deviation σ
e
of 4 me-
ters for DIST-DIFF, and a parameter λ
t
corresponding to a mean of 128 meters for OURS.
For DIST-DIFF, the value of σ
e
almost exactly matched the value suggested in [109],
which was 4.07 meters.
For OURS+SM, we multiplied the emission weights by w
N
= 1.5 if the scheduled
68 Chapter 2. Map-Matching Schedule Data
Table 2.6: A
10
N
scores (percentage of trips with an A
N
score ≤ 10%) for all evaluated
methods, without noise, on all datasets with ground truth data available. Best results in
bold.
G-STS DIST-DIFF OURS OURS+SM OURS+LM OURS+SM+LM
VG 18.4% 59.2% 73.1% 73.1% 73.1% 73.1%
SE 0.8% 92.8% 97.3% 97.3% 98.1% 98.1%
ZH 28.3% 84.5% 89.7% 89.9% 91.7% 92.0%
V 24.1% 97.9% 98.1% 97.9% 97.8% 97.6%
avg 17.9% 83.6% 89.5% 89.6% 90.2% 90.2%
station and the candidate station were not classied as similar. For OURS+LM, we added
a penalty factor of w
L
= 1.2 to edge weights with no matching line information on them.
OURS+SM+LM used exactly the same penalties.
The results for our quality evaluation without Gaussian noise can be found in Ta-
bles 2.6, 2.7. Tables 2.8, and 2.9 give the results under the same setup, but with a Gaussian
noise with standard deviation of 30 meters. All experiments were done on an Intel Xeon
E5-1650 machine with 6 cores, each with 3.5 GHz. Under Gaussian noise the results were
averaged from 50 runs. Our main results can be summarized as follows:
Baseline techniques are unsuitable for practice. The G-STS method, arguably the
most elaborate of our baseline techniques as it globally optimizes the path over
the station candidates, was clearly inferior on all datasets (without noise). Under
the δ
aF
metric, G-STS performed much better (but still worse than all other tested
methods), suggesting that G-STS mainly has a problem with correctly placing sta-
tions.
For precise input data, DIST-DIFF is already very good. Without noise, DIST-DIFF
performed very well, achieving similar results as OURS on the Vienna dataset. Un-
der the δ
aF
metric on Vienna, it outperformed all our approaches by a small margin.
DIST-DIFF is less robust against noise than our approach. Under noise, the qual-
ity of the DIST-DIFF decreases signicantly faster than our approach. This eect
can already be seen from Figure 2.26.
Penalties for unsimilar stations or lines help under noise. Without noise, both the
penalty for unmatched stations and unmatched lines improved the A
10
N
scores only
very little - for one dataset (Vienna), the results even got slightly worse. The δ
aF
improved for all datasets, however. Under noise, the penalties greatly improved the
results, from 6.3 percentage points for the Zurich dataset to 12.4 percentage points
for the Seattle dataset. δ
aF
improved similarly.
2.10 Experimental Evaluation 69
Table 2.7: δ
aF
scores (average Fréchet distance in meters), averaged over all trips. Best
results in bold.
G-STS DIST-DIFF OURS OURS+SM OURS+LM OURS+SM+LM
VG 6.2 8.5 4.4 4.4 3.5 3.5
SE 20.4 14.7 10.0 9.7 9.0 8.7
ZH 16.0 9.2 8.0 7.9 7.7 7.6
V 15.1 7.5 7.8 7.8 7.7 7.7
avg 14.4 9.9 7.5 7.5 7.0 6.9
Even with no noise, the baseline G-STS method was clearly inferior to all other
tested methods under the A
10
N
score, achieving only an average of 17.9% of trips with
an A
N
value of under 10%. Under the average Fréchet distance, the dierence was
less pronounced (Table 2.7), and for the Vitoria-Gasteiz dataset, it scored even better
than DIST-DIFF. This suggests that while G-STS is bad at placing stations correctly, the
overall vehicle paths are mostly correct. However, OURS, OURS+SM, OURS+LM, and
OURS+SM+LM, still clearly outperformed G-STS.
To good performance of DIST-DIFF with no noise under both the A
10
N
score and the
average Fréchet distance can be explained by the high quality of the evaluation datasets
and the aggressive emission probability distribution parameter σ
e
= 4. This results in
stations being almost always placed at exactly the position where the ground truth ex-
pects them, with the corresponding road segments being connected by the shortest path.
DIST-DIFF performed particularly well on datasets covering cities with a regular trans-
portation network. For example, both Seattle and Vienna have a grid-like road network
with many long straight roads. Here, the DIST-DIFF method shines, as it gives the high-
est probability to transitional paths with a length close to the sample point distance. This
was reected in the ground-truth data: the median dierence between the meter length
of the path connecting two stations, and their distance, was only around 1 meter (!)
for Seattle. For Vienna, it was around 7 meters. In comparison, it was over 12 meters
for Sydney, and around 19 meters for both Vitoria-Gasteiz and Zurich. (The DIST-DIFF
method was evaluated on a dataset from Seattle in the original publication [109]).
As G-STS adds no penalty to candidate stations with a larger distance from the sam-
ple station, its quality under both metrics decreased only little under Gaussian noise
(caused by the slightly dierent station candidate sets). The quality if the DIST-DIFF
approach decreased more quickly under noise than OURS. For a low standard devia-
tion of the emission probability distribution, this was to be expected, as the emission
probability largely denes the outcome. But even for higher standard deviations, the
quality under noise decreased faster than for OURS, as seen in Figure 2.26. Interestingly,
the quality decreased most for the Seattle dataset, with an A
10
N
score of only 9.9% under
70 Chapter 2. Map-Matching Schedule Data
Table 2.8: A
10
N
scores (percentage of trips with an A
N
score ≤ 10%) using the same
setup as in Table 2.6, but under Gaussian noise with a standard deviation of 30 meters
to simulate low-quality input data. Best results in bold.
G-STS DIST-DIFF OURS OURS+SM OURS+LM OURS+SM+LM
VG 15.5% 10.6% 49.3% 55.6% 55.7% 56.6%
SE 0.9% 9.9% 52.2% 55.0% 62.3% 64.6%
ZH 22.8% 34.8% 67.6% 67.0% 73.4% 73.9%
V 23.8% 51.0% 75.6% 77.9% 79.9% 82.6%
avg 15.8% 26.6% 61.2% 63.9% 67.8% 69.4%
Table 2.9: δ
aF
scores (average Fréchet distance in meters) under Gaussian noise with a
standard deviation of 30 meters. Best results in bold.
G-STS DIST-DIFF OURS OURS+SM OURS+LM OURS+SM+LM
VG 9.1 19.3 6.8 6.0 5.5 4.9
SE 20.1 30.9 14.1 13.9 11.6 11.4
ZH 16.4 18.2 10.7 10.8 9.6 9.5
V 15.5 16.2 10.4 10.4 9.2 9.2
avg 15.3 21.5 10.5 10.3 9.0 8.8
noise, compared to 92.8% with no noise.
In general, the additional penalty for unsimilar station candidates and unmatched
lines gave only moderate quality improvements without noise. On average, the A
10
N
score
under no nose improved only by 0.7 percentage points from OURS to OURS+SM+LM.
Under noise, however, it signicantly improved from 61.2% to 69.4% on average. This
of course depended on the quality of the corresponding metadata in OSM. For example,
both the line and station data in OSM was incomplete or seemed outdated for Seattle,
explaining the drop of the A
10
N
score for OURS+SM+LM from 98.1% under no noise, to
64.6% under noise. On the other hand, metadata quality was excellent for Vienna.
2.10.3 Speed
To nally measure the impact of our speedup techniques, we ran our approach with un-
matched station penalty and unmatched line penalty (OURS+SM+LM) with the parame-
ters from the previous section on all datasets (including those without ground truth data
available) using the following speedup techniques: the baseline technique which calcu-
lates all shortest paths for a single hop from H
i
to H
i+1
using |H
i
| 1-to-|H
i+1
| shortest path
2.10 Experimental Evaluation 71
Table 2.10: Running times of our baseline approach B (clustered equivalent trips and n
1-to-m Dijkstra runs between layers), CA (with shortest path cost caching), SDL (single
Dijkstra run between two layers), CA+TR (shortest path cost caching on a trip trie), and
SDL+TR (single Dijkstra run between layers on a trip trie). The unstarred versions use
a standard Dijkstra implementation, the starred versions use the A
∗
heuristic described
in Section 2.9.2. Best times are printed bold.
B CA SDL CA+TR SDL+TR B* CA* SDL* CA+TR* SDL+TR*
VG 2.3s 1.3s 0.1s 1.3s 90ms 1.9s 1.2s 0.1s 1.2s 83ms
SE 1.6m 1.4m 3.3s 1.3m 2.9s 57.5s 49.3s 2.2s 47.7s 1.9s
ZH 35.1s 26.3s 1.8s 25.1s 1.4s 31.1s 22.6s 1.7s 21.8s 1.3s
V 42.7s 35.2s 1.9s 30.7s 1.5s 32.4s 26.2s 1.7s 22.7s 1.4s
P 18.6m 14.0m 43.7s 13.7m 38.3s 10.2m 7.8m 27.4s 7.5m 23.7s
CH 1.5h 39.8m 2.8m 34.6m 2.2m 50.3m 21.2m 1.7m 17.3m 1.3m
DE 10.2h 7.6h 23.2m 7.5h 21.0m 3.8h 2.5h 9.9m 2.4h 8.5m
calculations with a basic implementation of Dijkstra’s algorithm (B), the baseline with
additional caching of shortest path costs (CA), the method for calculating all required
shortest path costs using a single Dijkstra run between hops (SDL), the cache based ap-
proach with better bulk processing using trip tries (CA+TR), and the single Dijkstra run
approach on trip tries (SDL+TR). For each method we evaluated a raw version, and a
starred version using the A
∗
heuristic described in Section 2.9.2. Note that the baseline
version was already only run on unique trips. We did not measure the time required to
parse the OSM input, as this parser is still preliminary and can so far only handle OSM
XML les.
The experiments were run on an Intel Xeon E5-1650 machine with 6 cores, each with
3.5 GHz. Results are reported in Table 2.10.
When compared to the baseline, an additional shortest path cost cache gave only
moderate improvements. This can mainly be explained by the relatively small overlaps
of unique trips in all test datasets. If a shortest path cost cache was combined with the
trip tries technique (CA+TR), the additional eect was only small as any additionally
calculation saved by a common prex of two trips would already have been saved by the
cache. The additional speedup of CA+TR when compared to the standalone CA tech-
nique can partly be explained by the reduced overhead of the cache bookkeeping. We
also only use one shortest path cost cache per thread to avoid the overhead of manag-
ing parallel writes to the cache, so some lost cross-thread caching opportunities were
salvaged by the trip trie preprocessing.
The eect of the SDL technique was dramatic, allowing us to map-match the entire
schedule data for Germany in under 25 minutes when used alone, compared to over 10
72 Chapter 2. Map-Matching Schedule Data
hours with the baseline method. An additional calculation of the trip tries (SDL+TR)
again only gave moderate improvements due to the relatively low overlap of unique
trips.
The A
∗
heuristic had a stronger impact on datasets covering a large geographical
area, bringing for example the baseline running time on Germany down from over 10
hours to only 3.8 hours. On smaller datasets the improvement was much smaller, as
expected in Section 2.9.2.
Using the SDL+TR* technique, we were able to completely map-match all test datasets
in under 9 minutes. For our largest testing dataset Germany, this meant a map-matching
time of around 2 milliseconds per unique trip, without any preprocessing besides the
building of the trip tries. We note that this time does not only include the map-matching
process, but also the building, simplication, and serialization to disk of the GTFS shapes.
Given that a naive baseline approach would require |H
i
| · |H
i+1
| shortest path calcula-
tions for a single hop (for the bus network of Germany, the average |H
i
| was 21.3, and
the average number of stops per trip was 20.9), and given the fact that we use dynamic
edge costs based on the routing attributes, we consider this a satisfying result.
2.11 Conclusions and Future Work
This chapter adapted a state-of-the-art map-matching approach using a hidden Markov
model to map-match schedule data, where sample points are only given at stations. We
described how turn restrictions can both be respected inside single transitions between
hidden states, and across hidden state transitions, by routing on the edge-to-node dual
graph, with vehicle orientations at candidate nodes encoded by adjacent edges. We
showed how metadata present in the underlying transportation network, in particular
station and public transit line labels, can be incorporated into this approach. In this con-
text, we evaluated several methods for deciding whether two station identiers, consis-
tent of a geographical position and a label, are similar. The quality of our raw approach,
and with the consideration of the network metadata, was compared to a simple baseline
method, and an established map-matching approach using a HMM, showing that our
approach already performs best without considering the network metadata. Our evalu-
ation also showed that the network metadata makes the approach more robust against
low-quality input data.
Using several speedup techniques, we were able to map-match the entire national
schedule data for Germany in 8.5 minutes. Smaller datasets for individual cities were
typically map-matched in under 2 seconds. It would nevertheless be interesting to eval-
uate faster algorithms for the shortest path calculation that lies at the core of our transi-
tion probability calculation. Using Contraction Hierarchies (CHs) here seems promising,
but a direct application requires a constant cost metric. As our transition probabilities
2.11 Conclusions and Future Work 73
A
B
c(B) = 30 m
c(A) = 60 m
H
1
H
2
Figure 2.27: Map matching a trip from a station
s
1
to s
2
, with candidate sets H
1
and H
2
. Path A
takes much longer (60 minutes) than path B (30
minutes). However, if the scheduled travel time
from s
1
to s
2
is around 60 minutes, A is the more
likely path. Our approach so far does not use this
temporal information.
use the vehicles line names to compute a dynamic edge cost function, a direct applica-
tion of CHs is not possible. However, techniques exist in the literature for using CHs
with a variable cost metric [47].
Our work so far does not use the temporal information present in public transit
schedules, but always prefers the route of shortest travel time. However, as Figure 2.27
shows, the fastest route between two station candidates may not necessarily be the most
likely one. We already experimented with approaches which based the transition proba-
bility on the dierence between the schedule travel time, and the real-world travel time,
but found the results to be vastly inferior. The obvious reason is that the travel time
given in schedule data is extremely unreliable: it is typically rounded to the nearest
minute (sometimes resulting in scheduled travel times of 0 between close stations), and
often includes buer times to improve punctuality. Additionally, the road travel times
extracted from the transportation network are only estimates, and may dier greatly
from the real-world travel times depending on the time of day or even the vehicle type
(a regional train, for example, travels a lot slower than a high-speed train).
Finally, our approach may also be a valuable pre-processing step for crowdsourcing
public transit vehicle information. Given the most likely path a vehicle will take between
stations through a network, it is straightforward to interpolate the approximate position
at time t from the schedule data. This would enable us to do map-matching on the
temporal domain: given the timestamped location measurements of a smartphone over
the last minutes, we would then ask for the most likely public transit vehicle the owner of
the smartphone is currently traveling in. Once a suciently probable coupling between
a smartphone and a public transit vehicle is established this way, the smartphone can
be used to provide a precise location of the vehicle. Together with the schedule data,
public transit vehicle delay information could thus be crowdsourced from passenger
smartphones.
74 Chapter 2. Map-Matching Schedule Data
Chapter 3
Line Graph Construction
The previous chapter described how geographical line courses can be extracted from
schedule data and a physical transportation network. This map-matching approach
yielded a bag of trips (S, L, Λ, T) and for each consecutive stop pair (s
i
, s
i+1
) of a trip
T ∈ T a polyline describing the geographical course between s
i
and s
i+1
. This already
has a natural graph representation tting the line graph denition from Section 1.3.2:
for each s
i
∈ S, add a node v
i
to the line graph G. For each consecutive stop pair (s
i
, s
i+1
)
of trip T , add an edge {v
i
,v
i+1
}, labeled with the trip’s line L and the hop geometry. This
line graph may be a multigraph, as there are typically hundreds of trips between two
stops. If all edges from v
i
to v
i+1
have exactly the same geometry (within some toler-
ance), it is straightforward to transform this multigraph into a graph - simply combine
all edges into a single edge labeled with (1) the shared geometry (or maybe the aver-
age geometry), and (2) the union of all edge lines. There are three problems with this
approach:
1. The line graph may be degenerate: if each (unique) trip in the bag of trip has distinct
stops, then G will be a collection of path graphs.
2. Two edges e and f connecting nodes u and v may have dierent geometries.
3. These might still overlap.
Consider Figure 3.1, left. Even if stations A, B, and C would be distinct nodes, prob-
lems 2 and 3 persist. If we solve problem 2 by merging the edges labeled with A and B,
and the edges labeled with D and C (Figure 3.1, middle), there would still be considerable
overlap in the nal map rendering (Figure 3.1, right). In this section, our aim is thus to
construct a so-called free line graph - suitable for rendering - from either a bag of single
line geometries (the degenerated case), or from an existing line (multi-) graph with edge
segment overlap. Figure 3.2 gives an example. We hence consider the following problem:
Problem 3 (Free Line Graph Construction). Given an arbitrary line (multi-) graph G =
(V , E, L, L) in which each edge is outtted with a polyline describing its course. Construct
a free line graph G
∗
from G which is suitable for rendering. In a free line graph, similar
75
76 Chapter 3. Line Graph Construction
{A}
{B }
{D }
{C }
Station B
Station C
Station A
v
w
u
{A, B}
{C, D }
Station B
Station C
Station A
Figure 3.1: Left: Degenerate line graph in which edges corresponds to unique trips.
Middle: Nearby nodes were rst merged, and the edges in the resulting multigraph col-
lapsed into single edges with averaged geometries. The resulting line graph does not
track shared segments. Right: Map rendered from this line graph with line overlap.
{A}
{B }
{D }
{C }
Station B
Station C
Station A
u
0
{A, B}
{C, D }
{A, B, C, D}
Figure 3.2: Left: Input line graph from Figure 3.1. Middle: Free line graph in which
shared segments are represented by distinct edges. A new node u
0
marks the point where
lines {A, B} and {C, D} branch. Right: Map rendered from this line graph.
stations should be represented by a single node (if possible), and edge segments overlapping
(within some threshold distance
ˆ
d) should be merged into a single new segment.
The remainder of this chapter is organized as follows: In Section 3.1, we discuss
related work. In Section 3.2, we describe a simple and fast method to construct a support
graph from the input graph G. Sections 3.3 and 3.4 then describe how we infer the
original line turn restrictions, and the (clustered) stations back into the support graph.
We give experimental results of our reference implementation in Section 3.5.
3.1 Related Work
Problem 3 can be considered a map construction problem, which is an active research
area. There, the goal is to automatically derive the underlying road network from a col-
lection of vehicle trajectories, typically GPS traces. One particular question that directly
translates to our setting is how the quality of constructed maps can be assessed. There
is also some overlap of our work with edge bundling.
3.1 Related Work 77
3.1.1 Map Construction
As most mobile phones today are outtted with GPS devices (or similar technologies
such as Galileo, BeiDou, or GLONASS), a vast source of GPS traces exists. These may
be used to construct maps for previously uncharted areas, augment maps constructed
from aerial imagery, or update existing road network data. In the past, these task were
typically performed manually. The advent of map services like Google Maps, Apple
Maps, Bing Maps, or TomTom has led to an increased research interest in recent years,
and hence several approaches to this problem have been developed. Ahmed et al. [1]
group existing approaches into three categories: (1) point clustering based approaches,
(2) approaches based on incremental construction, and (3) approaches based on intersec-
tion linking. These categories often blend, with many approaches using elements from
multiple categories.
Point clustering based approaches assume a set of points as the primary input (they
are typically obtained by using the raw input GPS points, or by sampling the input tra-
jectories) and infer the map between these points (the raw GPS traces are often used to
infer meta data, however). Already in 2003, Edelkamp and Schrödl [51] proposed to use
the k-means algorithm to cluster the input points into k seed nodes. Two seed nodes are
then connected if two points in the corresponding cluster were connected by an original
trace (in this regard, it uses elements from intersection linking based approaches). The
center line of the corresponding traces was used as an edge geometry. This approach
was later rened in [130]. In particular, lanes and intersections were carefully recon-
structed, and turn restrictions were considered by only connecting two reconstructed
lanes if an original trace connected them.
Other approaches rst transform the input traces into the black-and-white pixel do-
main and skeletonize the resulting image. For example, Davies et al. rst compute a
density grid [44] with each cell measuring the number of original input traces passing
through it. Afterwards, this grid is transformed into the pixel domain and the contours
of the edge segments are extracted. For the polygon edges, a Voronoi diagram is com-
puted, and the nodes of the corresponding Voronoi graph that are outside the contour
polygons are discarded. This results in a skeleton graph in which edges approximately
follow the road contour centerlines, further cleaned up by removing very short edges.
Input traces are then map-matched onto the skeleton graph to infer meta information.
Biagioni and Eriksson [28] use a similar approach, but smooth their 2D density grid
of 1 × 1 meter cells via a convolution with a normal distribution density function. The
resulting grid is then transformed into a black-and-white image using increasing thresh-
olds, and a skeleton graph is successively constructed. The original traces are then also
map-matched onto the skeleton graph, using a hidden Markov model. In this last step,
unused edges are discarded, and turn restrictions are inferred. For overviews over point-
clustering based map construction algorithms, see for example [27] or [1].
78 Chapter 3. Line Graph Construction
Incremental approaches begin with an empty map and iteratively add traces, typically
merging new traces with existing segments. The nal map is gradually constructed
in this fashion. An early example of this approach was presented by Rogers et al. in
1999 [127] (their approach always augmented an existing map): given some existing
map and a GPS trace t, sample both the map and the trace every 10 meters, merge each
input point with the nearest existing map sample point, and average the node position.
A similar approach was used by Cao and Krumm [35], but they rst mitigated GPS
noise by applying a force-based pre-processing step on the GPS traces: each GPS trace
point is attracted to its initial position, and to other trace segments, with a quadratically
decreasing force. This bundles traces belonging to similar road segments closer together.
Afterwards, the nal map is constructed in a similar fashion as in [127], but if no merge
node within a threshold distance is found, a new node is inserted. Our own approach
for constructing the support graph is heavily inspired by the work of Cao and Krumm.
Although we do not use a force-based approach for preprocessing, we rst run our con-
struction process with a very low threshold value to group nearby edge segments. Our
construction algorithm then proceeds similarly to their work, but employs some addi-
tional heuristics to prevent the merge of map segments meeting at large angles, and to
clear the map of merge artifacts.
Ahmed and Wenk [2] describe an incremental algorithm that is based on partial map-
matching: using a previously described map-matching approach [5], they rst nd in a
graph G embedded in R
2
(modeling the map constructed so far), and for a single polyg-
onal curve C (modeling the GPS trace) the matching graph segments that optimize the
overall Fréchet distance between the segments and C. For the partial map-matching part,
the free-space diagram [6] corresponding to the maximum allowed Fréchet distance is
established for each edge in G and C. These free-space diagrams are then glued together
according to the edge adjacencies, and a monotone path through the resulting free-space
surface that minimizes the usage of illegal (going through parts violating the Fréchet dis-
tance threshold) parts of the free-space surface is computed. The parts of the monotone
path that pass through illegal parts are the unmatched segments for C. For these, cor-
responding new graph edges are added. The geometries of the matched segments are
updated.
Intersection linking algorithms rst establish the intersection nodes in the corre-
sponding road network, and subsequently connect these nodes. Intersection nodes are
generally dened as points in the network where vehicles often change their direction.
As mentioned above, the k-means clustering based method described by Edelkamp and
Schrödl [51] can also be considered an intersection linking method.
Fathi and Krumm [56] propose a method based on machine learning. They rst dene
a local shape descriptor which is a circular window partitioned into annular segments.
This shape descriptor then samples the input traces and for each sample point counts for
each annular segment how many nearby traces are passing through it. The resulting cir-
3.1 Related Work 79
cular histogram is then transformed into a feature vector. A classier is trained to detect
whether a local environment described by a shape descriptor belongs to an intersection.
Given then a set of points that were classied as intersection nodes, they discard an in-
tersection node if there was another intersection node with higher condence within a
threshold radius. The nal road segments between intersection nodes are reconstructed
by using the geometry of the shortest connecting trace.
Karagiorgou and Pfoser also consider the speed information contained in the GPS
trace data [90] to extract intersections. Each GPS sample point is rst checked whether
it satises the turn condition. The latter is satised if the speed falls below 40 km/h, and
if the vehicle direction changes by at least 15
◦
. These so-called turn samples are then
clustered by their proximity into intersection nodes, using a threshold distance of 25
meters. The intersection nodes are then connected by averaging all trajectory segments
passing through them.
Some works focus exclusively on nding the road intersections. For example, Xie
et al. [154] describe a method for nding intersections by searching for the Longest
Common Subsequences (LCSS) of each pairwise GPS trace, using the start and endpoints
of a subsequence as intersection points. These are then further clustered using Kernel
Density Estimation.
Our own approach described in this chapter can be considered an incremental inser-
tion algorithm. The main dierence between our work and previous works are: (1) Our
input data is typically not a collection of traces, but may already be a line (multi-) graph
with many overlapping edge segments. (2) Our constructed line graph should contain
line information, that is, for each edge e, we must establish the set L(e) of lines travers-
ing through e. (3) We must cluster station nodes, and re-insert these clustered stations
at appropriate positions. (4) We do not aim at an exact geometrical reconstruction of
intersection lanes, but want to specically merge large intersections into single nodes,
as this allows for a cleaner rendering of the map later on. Nevertheless, (line) turn re-
strictions at these intersection nodes should be considered. (5) We do not have access to
speed information at individual sample points.
Before settling for the approach described in this chapter we experimented with
other techniques. In [19] and [20], we presented an approach where we swept each
input edge e and checked for other input edges within a threshold distance
ˆ
d. As soon as
such an edge f was found, we projected the position on e to f and marked the beginning
of a segment. This segment was then extended along e and f as long as the distance be-
tween them was lower than d (we allowed for short outliers). The matching segment of
e and f were then averaged, and nodes inserted at the beginning of the segment. Using
various heuristics and clean-up rules, we tried to both track the lines traveling through
the edges, the station positions, and the turn restrictions simultaneously. In practice,
this approach lead to many artifacts which had to be removed with subsequent heuristic
clean-up rules. It proved to be particularly dicult to correctly track turn restrictions
80 Chapter 3. Line Graph Construction
and to ensure that similar stations within the snap threshold were merged into a single
station node in the nal line graph.
3.1.2 Quality Measures
Early work on map-construction typically conducted a manual inspection of the con-
structed maps due to lack of machine-readable ground truth data. Over the years, sev-
eral quality measures for comparing a map-constructed graph to a ground truth graph
were developed. This problem is closely related to the general graph-theoretical prob-
lem of measuring the similarity between two graphs [42, 69, 65, 155], although spatial
graph embeddings are typically not considered there [1]. Hence, several distance mea-
sures between map-constructed graphs and a ground truth graph have been developed.
These range from simple approaches which densely sample both the constructed and the
ground truth graph and compare the resulting point sets (e.g. by measuring the Haus-
dor distance [7] or by computing specially developed measures [27, 103]) to path-based
measures which compare selected paths through both graphs. For the latter, a random
set of corresponding shortest paths is often used [108, 90].
3.1.3 Edge Bundling
There is also some overlap with our work and edge bundling, where the goal is to merge
edges to decrease visual clutter when drawing complex graphs. In contrast to our work,
where the original geographical line courses should be maintained, the edge embeddings
are typically not part of the input and can be freely chosen to create as many bundles as
possibles, while adhering to some optional side constraints. For example, Holten and van
Wijk [83] describe a force-directed approach in which edges attract each other to form
bundles. Pupyrev et al. consider edge bundling to produce graph drawings that look like
metro maps, also without considering input embeddings. They rst nd an edge-bundled
embedding which optimizes the total edge lengths and bundle widths. Afterwards, they
optimize the edge orderings on the shared segments using a restricted variant of the
metro-line crossing minimization problem (this will be discussed in Chapter 4).
3.2 Support Graph Construction
We rst describe how to build a support graph. A support graph is a line graph in which
all nodes are non-stations, and no turn restrictions are present. We build this support
graph incrementally: given some input line graph G, we order the edges by the length
of their geometries. These ordered edges e
1
, . . . , e
n
are then iteratively inserted into an
initially empty support graph H
0
, where H
i
is the support graph after the insertion of
3.2 Support Graph Construction 81
H
j
i−1
e
i
l
(1)
H
j
i−1
e
i
(2)
H
j
i
(3)
H
j∗
n
f
(4)
Figure 3.3: Constructing the support graph H
j
for an input graph G. The edges e
i
of G
are iteratively inserted into H
j
i−1
by rst sampling them and merging sample points with
the nearest existing node. Afterwards, degree 2 nodes are contracted where possible.
edge e
i
. For the insertion of edge e
i
, we densely sample the geometry of e
i
and then
merge sample points with the nearest existing nodes, if there are any within a threshold
distance. An additional step, artifact cleanup, will be described in the following section.
After all edges have been inserted into H, we start the process again with H as the input,
and continue this until a convergence criterion is met. The support graphs constructed
in reach round will be denoted as H
1
, . . . , H
m
.
Figure 3.3 gives an overview over the basic approach. We have an already existing
support graph H
j
i−1
from a previous iteration and want to insert edge e
i
. The edge geom-
etry of e
i
is rst sampled every l length units resulting in a list p
1
, . . . , p
o
of coordinates.
We then iterate over the coordinates and retrieve for each p
k
the nearest existing node
v in H
j
i−1
from a grid index. If the distance between v and p
k
is smaller than or equal
to a threshold parameter
ˆ
d, we average the position between v and p
k
, move v to the
resulting new coordinate, and set v
k
= v. If the distance between v and p
k
is greater
than
ˆ
d, we add a new node v
k
for p
k
. In both cases, an edge f = {v
k−1
,v
k
} is added
82 Chapter 3. Line Graph Construction
connecting the node of the previous sampling point to v
k
. We set L(f ) = L(e), that is f
receives exactly the lines traveling on the original edge e. In case the edge f = {v
k−1
,v
k
}
already existed, we set L(e) = L(e) ∪ L(f ). It is important to highlight that the edges in
the intermediate support graphs H
j
i
are not labeled with a polyline - their geometry is
completely dened by the node embedding.
Observe that a naive application of this process would subsample the geometry of
each input edge as we would always merge a sampling point p
k
with the previously
inserted node as long as the distance is below
ˆ
d. To avoid this, we keep a blocking set B
of nodes corresponding to the last
ˆ
d/l sampling points and never use a node v ∈ B as a
candidate node.
After all edges have been inserted, we contract degree 2 nodes in the resulting graph
H
j
n
if (1) the adjacent lines have exactly the same lines, and (2) the contraction would
not lead to a multigraph. During this process, the edges are re-labeled with polyline
geometries: the geometries of the adjacent edges д and h are concatenated and used
as the geometry for combined edge дh. This graph H
j∗
n
is then used as the input for a
subsequent iteration, or returned if the convergence criterion is met.
A major limitation of this approach is that matching segments of H
j
i−1
and e
i
may not
be completely merged. For example, in Figure 3.3, the blue edge f is left as an artifact.
The next section describes how to handle such issues.
3.2.1 Artifacts
The process described so far leaves two types of artifacts, both depicted in Figure 3.4.
Type 1 artifacts will be cleared by further iterations of our approach. A typical example
is the triangular structure depicted in Figure 3.4, top. Node u was not contracted at the
end of the rst iteration as it would have led to a multigraph. After node u was added
to the graph, another merge moved node v so that u is now within distance d <
ˆ
d to
e. If the length of e is above the sampling length l, a sampling point will be inserted
somewhere along e in the next iteration, at which point the edges adjacent to u will be
merged with the original edge e.
Type 2 artifacts typically involve one or more edges which are shorter than the sam-
pling length l and adjacent to nodes of degree > 2. Figure 3.4, bottom gives an example.
Here, edge e has a length below the sampling length l, which means that in a subse-
quent iteration, it will receive exactly 2 sampling points corresponding to the incidents
nodes u and v. As e is shorter than the edges f and д, it will be inserted rst into the
support graph of the next iteration. At this point, the sampling point of node v will be
snapped to the existing node at the same position, added during the construction of the
blue edges. Node u will be placed at exactly the same position, as edges f and д are not
yet inserted, and the sampling point corresponding to u cannot be snapped to the node
corresponding to v, as it is in the blocking set B. Edge e will therefore be reconstructed
3.2 Support Graph Construction 83
l
l
l
d
u
v
e
e
(1)
(2)
u
v
f
д
Figure 3.4: Two types of artifacts left by a single iteration of our approach. Top: If
triangular structures remain in which the edges are within the merge threshold (d ≤
ˆ
d),
and if an aected edge e has a length greater than the sampling length l, an additional
sampling node will be introduced in the next iteration, and the structure will collapse.
Bottom: The edge e is shorter than the sampling length and adjacent to nodes of degree >
2. It will be reproduced exactly by following iterations of our approach. We immediately
contract such edges after each iteration in a cleanup step.
exactly. But if the length of e was below the merge threshold
ˆ
d, we have missed a merge
opportunity, and we will miss it in each subsequent iteration.
To avoid such issues, we contract edges adjacent to two nodes with a degree > 2
and which have a length below l immediately after the insertion of an edge e
i
during the
construction process. As two or more edges may form a single edge with length < l after
the nal degree 2 node contraction, we additionally clear such edges in the iteration’s
nal degree-2-contracted graph H
j∗
n
.
3.2.2 Line Creep and Network Planarity Issues
A more subtle problem with the approach above is shown in Figure 3.5. Let e and f
be two adjacent edges in the input graph G, with L(e) = {A}, and L(f ) = {B}, whose
geometries meet at an angle close to 180
◦
.
Figure 3.5, left, shows a support graph into which f has already been inserted, and
edge e should now be inserted. If e is now sampled as described above and the resulting
sampling points snapped to H from left to right, the rst sampling point of e which is
in reach of the existing node u will be merged with u, and the subsequent sampling
points will be merged with nodes corresponding to sampling points on the original edge
f after u (again from left to right). Figure 3.5, middle, highlights the merged sampling
84 Chapter 3. Line Graph Construction
{𝐴}
{𝐵 }
𝑒
𝑢
{𝐴}
{𝐵 }
𝑒
{𝐴}
{𝐵 }
𝑢
0
𝑢
00
{𝐴, 𝐵 }
ℎ
Figure 3.5: Left: The input graph consisted of two edges e and f meeting at an obtuse
angle, f has already been inserted, e is to be inserted. Middle: The input edges will be
interlaced during the support graph construction. Right: After degree-2 node contrac-
tion, a segment with lines A and B appeared. This line creep will continue with each
iteration.
{𝐴}
{𝐵 }
𝑒
𝑢
{𝐴}
{𝐵 }
𝑒
𝑢
Figure 3.6: Left: To solve the line creep issue shown in Figure 3.5, we dene a no-merge
zone for sampling points close to the rst sampling point p
1
, and the last sampling point
p
o
of e: if for any other sampling point p
i
the distance to p
1
or p
o
is smaller than to some
merge candidate, we discard that candidate. Right: The resulting support graph.
points and existing nodes in light blue. This interlacing will lead to edges between the
merged nodes to share the lines of e and f , resulting in a segment in which both lines
are present (Figure 3.5, right). This will continue with each additional iteration until the
original input edges e and f are merged into a single edge h with L(h) = L(e)∪L(f ). We
call this eect line creep.
To mitigate this issue, we add another rule to the node candidate selection process
for the sample points: let e = {v, w} again be the edge to be inserted into a support
graph H, and let p
1
and p
o
be again the rst and last sampling point of e (corresponding
to the position of v and w). Let now p
i
with 1 < i < l be a sample point to be merged
into H . If a node v in H at position p
v
is within the merge radius of p
i
, we only merge
if α · dist(p
i
, p
1
) > dist(p
i
, p
v
) and α · dist(p
i
, p
o
) > dist(p
i
, p
v
) (p
i
is - within a factor α -
nearer to v then to either p
1
or p
o
). This eectively creates no-merge zones for sample
points close to p
1
and p
o
, as shown in Figure 3.6, left. We set α to sin(π /4) = 1/
√
2. The
intuition behind this is simple: if we assume that e.g. p
1
, p
i
and p
v
form a right triangle,
with the hypotenuse
←−→
p
1
, p
i
, then the condition is met as soon as
←−→
p
i
, p
1
and
←−→
p
i
, p
v
meet at
an angle of less than 45
◦
. It is easy to see that this prevents the merge of adjacent edges
which meet at an angle greater than or equal to approximately 45
◦
.
Another problem is that our approach so far tends to create complicated merge pat-
terns between edges which cross in the input graph G, and which also cross in the “cor-
rect” free line graph we are trying to construct. Figure 3.7 gives an example. The input
graph is not planar, and it is easy to see that there is no reason to merge the crossing
edges e and f , as they should also cross in the nal rendered map. As the edges cross in
3.2 Support Graph Construction 85
e
f
Figure 3.7: Left: Nonplanar input graph G where two edges e and f meet at an ap-
proximately 90
◦
angle. Middle, right: During the support graph construction, an artifact
merge zone between e and f will appear.
e
f
e
0
f
0
f
00
e
00
Figure 3.8: To avoid the merge zone artifacts shown in Figure 3.7, we rst build a support
graph using a very low merge threshold level (2l), which will planarize such intersection.
The no-merge zones will prevent the edges e
0
, e
00
, f
0
, and f
00
from being merged if the
resulting support graph H is used as input for a construction with higher
ˆ
d.
the middle, the no-merge zones described in the previous paragraph do not apply.
We avoid this with a simple preprocessing step: before constructing the support
graph for G with merge threshold
ˆ
d, we rst construct a support graph H with a very
small
ˆ
d (in our experiments, we used
ˆ
d = 2l). This eectively planarizes the input graph,
as shown in Figure 3.8. If H is now used as an input for a support graph construction
with threshold
ˆ
d, the original edges e and f have been broken up, and the resulting edges
e
0
, e
00
, f
0
, and f
00
meet at a common intersection node. At this node, the no-merge zones
prevent a further merge if e and f crossed at an angle greater than 45
◦
.
3.2.3 Convergence Criterion
As mentioned above, we apply the construction process in rounds, taking the nal sup-
port graph H
j−1∗
n
as input for the construction of a support graph H
j∗
n
. Several conver-
gence criteria come to mind for this process, for example the number of lines, the number
of edges, or the number of performed merges. During our experiments, we found the
86 Chapter 3. Line Graph Construction
Figure 3.9: Part of the input line graph for the Zurich tram network (green), and the
constructed free line graph (black). Edge geometries tend to be unsteady around large
intersections. We remove the parts within a distance of
ˆ
d around the stations and replace
it by a straight line.
edge length gap |1 −
L
∗
L
| to work best. Here, L is the sum of all edge lengths in the input
graph, and L
∗
is the sum of all edge lengths in the constructed support graph. In our
experiments, we stopped the iterative process if the edge length gap fell below 0.2%.
3.2.4 Intersection Smoothing
Despite the artifact cleanup measures described above, the support graph construction
process sometimes tends to slightly distort segment geometries at intersection nodes
(see Figure 3.9, left for an example). To later achieve a clean look in the nal drawing,
we apply a nal cleanup step: at each node v in the nal support graph, we crop
ˆ
d length
units of the geometry of each adjacent edge, set the position of v to the average of the
resulting end points of the adjacent edges, and connect each edge with a straight line
geometry to v. Figure 3.9 gives an example of this process.
3.3 Inferring Line Turn Restrictions
Section 1.3.2 already introduced line turn restrictions in line graphs, but so far we didn’t
motivate them in detail. As mentioned in the introduction, line turn restrictions are im-
portant if lines don’t follow simple paths through the network. Consider Figure 3.10,
which shows our running example of the Chicago loop rendered with considering line
turn restrictions (left), and without (right). Without them, the map shows several im-
possible turns for the blue, the brown, the light pink, and the orange line.
Previous work tried to track line turn restrictions through the support graph con-
struction process [19], but this turned out to be very error-prone. In this section, we
describe an alternative approach - instead of tracking the restrictions through the con-
3.3 Inferring Line Turn Restrictions 87
Figure 3.10: Left: Chicago loop, rendered with line turn restrictions considered. Right:
Same map, rendered without considering line turn restrictions.
struction process, we infer them onto the nal support graph. At rst glance, it might
seem enough to track the original edges merged into support graph edges during the
construction process and determine the turn restrictions based solely on edge adjacen-
cies. For an edge h in a support graph H , let M(h) be the set of original input edges
merged into h during the construction of H. If then, for example, input edges e and f
were adjacent in G, and we now check two edges h and i with e ∈ M(h) and f ∈ M(i),
then if some line l continued from e to f in the original input graph, we would deter-
mine that l also continues from h to i. However, depending on the merge order of input
edges, overlapping input edges may lead to an input edge e appearing in M(h) although
it only shared a short segment (see Figure 3.11 for an example of how that would dis-
tort line turn restriction based on the adjacencies of the M(e)). We use a more robust
method of comparing the lengths of shortest paths connecting special edge handle nodes
in the support graph to the shortest path lengths between corresponding handles in the
original input graph.
3.3.1 Edge Handles and Shortest Path Length Comparison
Consider Figure 3.12. Given a support graph H constructed from G, we rst add edge
handles to each edge in H . These are two nodes inserted at a progression of 1/3 and 2/3
on the corresponding edge geometry (Figure 3.12.2). This results in a support graph H
0
with edge handle nodes. For each pair of original support graph edges e and f adjacent
at a node u, we now have a pair of handle nodes α, β, where α is the handle node on e
which is closest to u, and β is the handle node on f which is closest to u.
88 Chapter 3. Line Graph Construction
д
e
d
f
{A}
{A}
{A}
{A}
(1)
u
(2)
{A}
{A}
{A}
M(h) = {d}
M(i) = {f , e}
M(j) = {e, д}
j
i
h
(3)
Figure 3.11: Turn restrictions based on adjacencies of merged edges distorted during
the support graph construction. (1) Input graph G. (2) After edges e, д, d, and f (in that
order) have been inserted into H, edges stemming from d and д are now adjacent at node
u. The next iteration will connect the edge stemming from f with an edge stemming
from e. (3) After degree-2 node contraction, the adjacencies of the merged original edges
M(h), M(i), and M(j) incorrectly infer a line turn restriction from h to j for line M, and
no line turn restriction from h to i.
In a second step, we draw an imaginary line (orthogonal to the corresponding sup-
port graph edge) of length a
ˆ
d (our merge threshold times some padding factor a) through
and centered at each handle node and add corresponding handle nodes to the original
input graph where edges intersect with the line (Figures 3.12.2 and 3). The padding fac-
tor a is required as the distance between a handle point on e and one of the original
edges merged into e might be greater than
ˆ
d, depending on the merge order. This results
in a line graph G
0
with edge handle nodes. We now have for each handle node α in the
support graph H
0
a corresponding set of handle nodes A in the original input graph G.
3.3.2 Shortest Path Comparison
For each pair α, β of edge handles in H
0
, we compute the shortest path from α to β
through the support graph H
0
(with edge weights based on the edge lengths), and store
its cost. We then also compute the cost of the set-to-set shortest path from A to B
through G
0
, but this time separately for each line l ∈ L(e) ∩ L(f ). In particular, we
set the weight of an edge e in G
0
to ∞ if l < L(e). Importantly, we now also consider
turn restrictions present in the original input line graph G, using the same approach as
described in Section 2.6.4 (an edge-to-node dual graph). If for two edges e and f (adjacent
at u) and some line l, the dierence between the shortest path cost between their edge
handles α, β through H
0
and the shortest path cost under l from A to B through G
0
is greater than a threshold t, we add a turn restriction from e to f at u for l. In our
experiments, we used a threshold t of 500 meters.
For example, in Figure 3.12.4, the shortest path through G
0
from A to B for the red
line A would take the entire loop, while the shortest path from α to β in H
0
would be
much shorter and go through node u.
3.3 Inferring Line Turn Restrictions 89
{A, B}
{B, C}
{A, B}
{A, B}
{A, B, C}
{C}
e
f
u
(1)
{A, B}
{B, C}
{A, B}
{A, B}
{A, B, C}
{C}
e
f
u
?
(2)
α
β
{A, B}
{B, C}
{A, B}
{A, B}
{A, B, C}
{C}
e
f
u
(3)
α
β
?
?
(4)
A
B
Figure 3.12: Adding turn restrictions onto a support graph H. Given a support graph
H for an original input graph G (1), we rst construct a support graph H
0
in which for
each edge two handle nodes are inserted at a progression of 1/3 and 2/3 of the edge
geometry. An imaginary line is then drawn through edge handle nodes, orthogonal to
the corresponding edge, and a set of handle nodes is added to the edges in G intersecting
this imaginary line. For two edges e and f adjacent at u, we compare the shortest path
connecting the handle points closest to u with the shortest path from the corresponding
set of handle nodes added to G, for each line l. If the shortest path cost dierence is
above a threshold t, we add a turn restriction for l from e to f at u.
90 Chapter 3. Line Graph Construction
3.3.3 Avoiding False Positives
Depending on the size of the padding factor a and the position of the imaginary cutting
line, we might add a handle point for an edge e in H to an edge f in G which wasn’t
merged into e during the support graph construction process. To avoid such false posi-
tives, and to also speed up the selection of appropriate edges, we maintain for each edge
e in the support graph H a set M(e) of original input edges that have been merged into
e, just like we keep track of the lines on e through the construction process. If the imag-
inary cutting for a handle point on support graph edge e line intersects an edge f in the
input graph, but f < M(e), we do not add an edge handle node there.
3.4 Station Clustering and Insertion
It now remains to show how the original stations can be re-inserted into the support
graph in a clustered way. We do this by selecting candidate insertion edges for each
station cluster, ordering them by a score, and inserting a node into the support graph at
the best candidate position.
Let S be the set of station nodes in the original input line graph G, and let each
station node s ∈ S be labeled by a function n which assigns s a station label (for example,
n(s) = Main Station). Observe that each s is served by a set of edges in G, namely adj(s),
and recall that we have kept a set M(e) for each edge e in the support graph H which
holds all original input edges merged into e during the support graph construction. As
we desire close similar stations to be represented by a single node in the nal map, we
rst partition S into subsets of similar stations. Let P be this partition. To decide whether
some s ∈ S belongs to a subset A ∈ P, we use a P + JAC similarity classier as described
in Section 2.8 and put s into A if ∃a ∈ A : sim
JAC+P
(s, a) > 0.5. Note that this implies that
no two subsets A, B ∈ P share a similar station. Additionally, we keep for each station
cluster A a set Q
A
of edges served by the s ∈ A, that is Q
A
=
a∈A
adj(a).
For each A, we then retrieve a set of edges in H that are within a threshold distance
r around the centroid of the a ∈ A. For each such retrieved edge e, we add 3 candidate
positions to the candidate set: the centroid projected onto the geometry of e, and the
position of the two adjacent nodes of e. Each candidate c then receives a score based on
(1) the distance d from the centroid of A to c, and (2) the number C of correctly served
edges at c. If c was the projected coordinate on e, then C = |Q
A
∩ M(e)|. If c was a node
v, then C = |Q
A
∩
e∈adj(v)
M(e)|. Based on these values, we calculate a score
o = d +
C
Q
A
· 100 (3.1)
and insert the station cluster at the candidate of highest score.
3.5 Experimental Evaluation 91
3.5 Experimental Evaluation
This section briey discusses the experimental results as well as existing problems (and
possible solutions) of our approach. We conducted a quantitative and a qualitative eval-
uation on 6 schedule datasets: a simple tram network with few necessary line turn re-
strictions (Freiburg-Tram), a medium light rail network with several necessary line turn
restrictions in which segments often meet at rectilinear angles (Chicago, thus testing the
eectiveness of our no-merge zones technique described above), a large and complicated
subway network in which many line courses cross (New York), an equally complicated
tram network requiring many line turn restrictions (Zurich-Tram), as well as medium-
sized bus network (Freiburg-Bus) and a large bus network (Zurich-Bus). The Zurich
datasets were also chosen because the ocial schedule dataset has imprecise shape data,
leading to large edge bundles on each network segment (as shown in Figure 3.19).
3.5.1 Experimental Setup
Preliminary line graphs were rst extracted from the schedule datasets for each of the
test networks using a straightforward approach: each stop in the GTFS dataset yielded a
node, and line connections between stops yielded a single edge, labeled with a geometry
that was extracted from the GTFS shapes. If no shapes were available, we rst map-
matched the dataset using our approach from Chapter 2 (this was only necessary for
the Freiburg dataset). From these preliminary line graphs, we then constructed a free
line graph using the method described in this chapter. As a segment merge threshold, we
used
ˆ
d = 50m. For a smoother look, we also applied a single round of Chaikin smoothing
to the nal line graphs [36]. The preliminary line graphs extracted from the schedule
data as well as the evaluation setup are available online
1
.
For the quantitative evaluation, we used the preliminary line graphs as ground truth
and compared them to the constructed nal line graphs using a shortest path-based
distance measure. We rst densely sampled both the ground truth graph and the test
graph. Then, 10,000 random node pairs (u, v) were picked from the ground truth graph.
For both u and v, we then picked all nodes within a distance of 3
ˆ
d, for both the ground
truth and test dataset. This yielded corresponding context sets U
G
and V
G
in the ground
truth graph, and U
T
and V
T
in the test graph. Afterwards, we computed the shortest set-
to-set path p
G
connecting U
G
and V
G
through the ground truth graph, and the shortest
set-to-set path p
T
connecting U
T
andV
T
through the test graph, both for a random line l ∈
L. The resulting paths were then compared using the Fréchet distance. This measured
(1) the topological correctness of the constructed line graph edges (and the lines traveling
on them), (2) the quality of the line graph edge geometries, and (3) the correctness of the
inferred turn restrictions. To also measure the quality of the re-inserted stations, we
1
https://github.com/ad-freiburg/topo-eval
92 Chapter 3. Line Graph Construction
Table 3.1: Quality of our line graph construction approach. We took 10,000 pairs of
random sample nodes and calculated the shortest path connecting the corresponding
context sets (all nodes within 3
ˆ
d meters around the sample node in the input line graph
and the constructed free line graph). The experiment was repeated for a set of random
station pairs. Under acc. we give the percentage of paths with a Fréchet distance under
3
ˆ
d. Under δ
F
min, max, and avg., we give the minimum, maximum, and average Fréchet
distance over all paths.
Random sample pairs Random station pairs
acc. δ
F
min δ
F
max δ
F
avg. acc. δ
F
min δ
F
max δ
F
avg.
Freiburg-Tram 98% 0.6 282.4 23.1 90% 0.63 159.9 42.8
Chicago 99% 0.2 1481.3 26.9 88% 0.1 1410.3 24.6
New York 95% 0.1 2575.5 40.9 61% 0 18682.9 150.4
Zurich-Tram 81% 1.2 4441.7 68.6 78% 1.9 2660.1 104.1
Freiburg-Bus 74% 1.1 2881.1 73.3 56% 1.8 3783.5 133.3
Zurich-Bus 71% 3.8 1486.4 103.9 61% 2.8 4212.1 166.9
conducted a similar experiment where we did not use random sample nodes for the
context sets, but random station pairs from the ground truth data. The context sets were
then stations from the ground truth and the test data that were similar.
We also measured the time and number of iterations required until convergence, the
time needed for inferring turn restrictions, and the time needed to re-insert the clustered
stop positions.
3.5.2 Results and Discussion
The results of our quantitative evaluation are given in Tables 3.1 and 3.2. The resulting
free line graphs for the Freiburg-Tram, Chicago, New York, and Zurich-Tram datasets
are also shown at the end of this chapter in Figures 3.15, Figures 3.16, Figures 3.17, and
Figures 3.18. Figure 3.19 additionally shows an excerpt of the Zurich city center area.
Table 3.1 gives the minimum, maximum, and average Fréchet distances between the
shortest paths. We also computed an accuracy score, which is the percentage of paths
with a Fréchet distance under 3
ˆ
d. Under this measure, and for random sample pairs, we
found the quality for Freiburg-Tram, Chicago, and New-York to be near-perfect, all with
an accuracy near 100%. For Zurich-Tram, the accuracy was 81%. Under manual investi-
gation, the main culprit for this lower score was an incorrect line turn restriction near
the station Stauacher, which was caused by a loop-like edge geometry in the prelimi-
nary line graph extracted from the GTFS data. For the Freiburg and Zurich Bus network,
3.5 Experimental Evaluation 93
Figure 3.13: Excerpt from the free
line graph of Zurich around the
Bellevue station. A single station
node was inserted for the clus-
tered Bellevue station, which is
not adjacent to line 9 which served
Bellevue station in the input line
graph, and a turn restriction pre-
vents line 8 from serving the sta-
tion. It is not possible to insert a
single station in such a way that
all lines serve the station.
the reasons for the lower accuracy scores of 74% and 71% were also often problems with
the inferred turn restrictions, but there were also many cases in which the Fréchet dis-
tances between shortest paths was just slightly above 3
ˆ
d, suggesting that the resulting
geometry was correct, but a series of edge merges has moved the nal edge geometry
more than 3
ˆ
d from an original input edge geometry (this often happened around sta-
tions covering a large area which were iteratively merged into a single node, with edge
geometries further distorted by the intersection smoothing described above).
For random station pairs, the accuracy scores were signicantly lower. The main
reason for this is that we are currently only inserting a single merged station node for
a cluster of similar stations. For large stations, however, it is sometimes impossible to
place a single node in such a way that all the edges (and therefore lines) originally serving
the station cluster are still serving the merged station node in the nal free line graph.
Figure 3.13 gives the example of the Zurich Bellevue station, where line 9 does not serve
the inserted station node in the nal line graph, and turn restrictions prevent line 8 from
serving it. Here, the station intersection should either be collapsed into a single degree-4
node, or an additional station node should be inserted.
To measure the general quality of the edge geometries, we did a careful manual inves-
tigation and noticed 5 kinds of common remaining problems over all datasets, as shown
in Figure 3.14:
S1. Edge geometries are sometimes not smooth. Depending on the order in which
nodes are merged, the resulting nal polyline may appear unsteady.
S2. Depending on the merge order of edges, intersection nodes may be moved towards
a certain segment.
S3. Intersection smoothing is at the moment also applied to nodes of degree 2. This
sometimes removes geographical details within the merge threshold around such
nodes (see the Wipkingerplatz station node in Figure 3.14).
94 Chapter 3. Line Graph Construction
S2
S1
S3
S4
S5
Figure 3.14: Excerpt from the Zurich tram line graph highlighting remaining problems
of our approach: (S1) Unsteady edge geometries. (S2) Lost geographical details around
degree 2 nodes. (S3) Displaced intersection nodes. (S4) Outlier trips may distort the line
graph. (S5) Intersection smoothing might add new merge opportunities.
S4. Infrequent trips with extreme course deviations may distort the line graph. In the
example, the outliers are vehicles returning to a depot at Escher-Wyss-Platz in the
evening.
S5. Intersection smoothing might introduce additional merging opportunities.
Additional smoothing after each iteration, each merge step, or after the entire sup-
port graph construction might be able to mitigate problem 1. A root cause for it (and
also for problem S2) is that the input edges so far carry no weight. If an edge e in the
support graph is already the results of 5 merges, and we are now merging a sixth edge f
onto it, then the new positions of the merged nodes will again be the average positions
of the node belonging to e, and the node belonging to f . It might be better to weight the
positions by the number of already merged edges.
Regarding problem S3, the obvious solution is to reconstruct intersections only for
nodes of degree greater than 3. Problem S4 is again related to problems S2 and S3, but
this problem suggests a weighting of the edges not by the number of original edges (and
thus the unique trips with the same shape on this edge), but by the total number of trips
traveling through this edge in the input schedule data (either in total, or on an average
weekday). It might also be sensible to consider completely dropping line connections
from the input data on which the total number of trips on an average weekday is below
a certain threshold (for example below 10% of the average number of line connections
in the entire dataset), as such trips are usually not depicted in ocial network maps.
The number of iterations until convergence as well as the running time, the number
of inferred restrictions, the maximum number of input edges merged into a single edge,
and the overall edge lengths in the input and free line graph are given in Table 3.2. In
3.6 Conclusions and Future Work 95
Table 3.2: Under it, we give the total number of support graph construction iterations
until convergence. t
|H |
is the time required for the support graph construction, t
R
the
time required for the inferring of line turn restrictions, and t
S
the time required for the
station insertion. The number of inferred line turn restrictions is given as |R|. K is the
maximum number of input edges merged into a single free line graph edge, and l
bef
and
l
aft
is the total summed edge length of the input graph (l
bef
) and the constructed free line
graph (l
aft
).
it t
|H |
t
R
t
S
|R| K l
bef
l
aft
Freiburg-Tram 6 1.1s 9ms 7ms 4 27 115.9k 52.2k
Chicago 4 3.6s 30ms 155ms 5 22 490.1k 220.3k
New York 5 7.2s 80ms 57ms 12 29 679.1k 442.9k
Zurich-Tram 15 11.2s 50ms 47ms 95 78 345.9k 125.8k
Freiburg-Bus 7 10.9s 136ms 43ms 238 61 930.7k 294.4k
Zurich-Bus 15 1.7m 1.1s 0.5s 1k 64 6741.1k 2496.3k
general, our approach quickly converged in under 16 iterations for all test datasets, and
a line graph was constructed in under 12 seconds for all datasets except the Zurich bus
network (here, the summed length of the input geometries was nearly 6,700 kilometers).
We are condent, however, that the running time can be further reduced with additional
engineering eort, for example by restricting the number of merge candidates for each
inserted sample point.
The total lengths of the nal free line graphs also seem to closely match the ocial
network lengths reported by the transit companies. For example, for the Freiburg tram
network, the summed edge geometry lengths were 52.2 km, and the ocial network
length is 43.9 km
2
. For the Zurich tram network, these lengths were 125.8 km, and 125.1,
respectively
3
. For the bus networks, the deviations were signicantly greater (294.4 km
vs 173.1 km for Freiburg, 2496.3 km vs 408.8 km). However, we note that the ocial
numbers were only for the single main urban bus company and the schedule data also
contained many regional lines. The ocial numbers also usually do not count outlier
lines or lines that use a slightly dierent course in dierent directions. These numbers
were thus hardly comparable.
3.6 Conclusions and Future Work
This chapter presented a method to construct a free line graph which can be rendered
as a transit map without edge overlaps from raw schedule data, or from a line graph in
2
https://de.wikipedia.org/wiki/Freiburger_Verkehrs_AG
3
https://de.wikipedia.org/wiki/Verkehrsbetriebe_Z%C3%BCrich
96 Chapter 3. Line Graph Construction
which such overlaps would be present (for example, a line graph directly extracted from
OSM data), a task that is crucial for the nal visual quality of the maps. To achieve this,
we rst constructed a support graph without any station nodes by iteratively inserting
edges from the input data into the initially empty support graph, merging nodes within a
threshold distance
ˆ
d. We used several heuristics to stabilize this approach. We described
a method to infer line turn restrictions present in the original input data onto this sup-
port graph, and described how the stations can be re-inserted into this support graph
in a clustered manner. For the clustering process, we used the station similarity classi-
er from the previous chapter. We conducted an extensive quantitative and qualitative
evaluation.
Although we still consider the results of this chapter preliminary, they already pro-
duced very good results, with remaining problems regarding the quality of individual
segment shapes, and the placement of the clustered stations. For the former, we see
great potential in weighting the sample points by the number of edges (or trips) already
merged into the edge they were sampled from, and considering this weight when aver-
aging the new position. For the latter, it might be necessary to insert more than a single
node into the nal free line graph for a station cluster.
It would also be interesting to evaluate our approach not on input data extracted
from schedule datasets, but from geographical datasets like OSM.
We note that our methods are not restricted to the domain of line graphs for transit
map rendering, but can in theory be applied to arbitrary graph-like input data. In par-
ticular, our method should also work for inferring turn restrictions onto road networks
constructed from vehicle GPS traces.
3.6 Conclusions and Future Work 97
Figure 3.15: Line graph for the 2022 tram network of Freiburg, created from raw sched-
ule data using the approach described in this chapter. The input schedule data was
rst map-matched with our approach from Chapter 2. The merge threshold was set
to
ˆ
d = 40 m. Stations are given in blue, non-station nodes in green, and nodes with a red
outline have inferred turn restrictions. Not all station and line labels are shown.
98 Chapter 3. Line Graph Construction
Figure 3.16: Excerpt from the line graph for the 2020 ’L’ network of Chicago, created
from raw schedule data using the approach described in this chapter. The merge thresh-
old was set to
ˆ
d = 40 m.
3.6 Conclusions and Future Work 99
Figure 3.17: Excerpt from the line graph for the 2018 subway network of New York,
created from raw schedule data using the approach described in this chapter. The merge
threshold was set to
ˆ
d = 40 m.
100 Chapter 3. Line Graph Construction
Figure 3.18: Line graph for the 2021 tram network of Zurich, created from raw schedule
data using the approach described in this chapter. The merge threshold was set to
ˆ
d =
40 m.
102 Chapter 3. Line Graph Construction
Chapter 4
Line Ordering Optimization
Our pipeline so far has produced an undirected line graph G = (V , E, L, L), where for
each e ∈ E, L(e) labels the edge with a set of transit lines passing through e. While we
may already render this line graph as a transit map, the result will be neither esthetically
pleasing nor informative (see Figure 4.1 for an example). Because L(e) is an unordered
set, it is not clear in which order the lines on e should be rendered. If we just use a
random ordering (or the input ordering) it will produce many situations where transit
lines cross or separate, making it very hard to follow line courses.
This chapter formalizes the problem of nding permutations (we say a line ordering)
of the L(e)that are informative. We describe and evaluate several algorithmic approaches
to nd such an optimal ordering very fast. In particular, we would like to have as few
undesired line crossings as possible and therefore consider the following problem:
Problem 4 (Optimal Line Permutation). Given a line graph G = (V , E, L, L). For each
edge e ∈ E, nd a permutation of the L(e) such that the readability of the nal map is
optimized. We call such a permutation a line ordering, and the set σ of all line orderings
for E a line ordering solution.
We will specify what constitutes optimal readability later on.
Our contributions in this chapter are the following: We give a formulation of the
Metro Line Crossing Minimization problem (MLCM) [26], which we call the Metro Line
Node Crossing Minimization problem (MLNCM). This formulation better matches real-
world transit maps and also simplies the map rendering process (a similar formulation
has previously appeared in the context of wire routing for electronic designs [75] and
edge bundling [123]). We then introduce a weighted version (MLNCM-W) which allows
specifying line crossing weights per node and line pair. We also introduce the novel
concept of minimizing line separations and formulate the MLNCM problem with count-
ing line separations (MLNCM-S) and with counting weighted line separations (MLNCM-
WS). MLNCM-WS will be shown to be able to produce line orderings very similar or
103
104 Chapter 4. Line Ordering Optimization
Figure 4.1: Left: The “Loop” of the Chicago light rail system with the unoptimized input
line ordering; it is very hard to follow lines. Middle: Same area, with an imperfect line
ordering (obtained via a local search approach). Right: Optimal line ordering.
equal to manually created line orderings in ocial maps. The concept of line separa-
tions partly replaces previous approaches to solve the problem of block crossings (this
will be further explained below). We give NP-hardness proofs for MLNCM, MLNCM-W
and MLNCM-WS (the NP-hardness of both MLNCM-W and MLNCM-WS directly fol-
lows from the NP-hardness of MLCM, however, in its raw MLNCM version, the problem
seems to defy a straightforward reduction proof from MLCM). On the positive side, we
describe a linear-time algorithm to solve MLNCM to optimality on line graphs where
each line follows a simple path and ends at a node of degree 1. We also give a fast
ILP formulation to optimize MLNCM-WS (and thus MLNCM, MLNCM-W, and MLNCM-
S), orders of magnitude faster than a simple baseline formulation using state-of-the-art
ILP solvers. Additionally, we describe and evaluate several heuristic approximation ap-
proaches to this problem. We also describe a set of pruning, cutting and untangling
rules which may be applied to the input line graph to (sometimes dramatically) reduce
the search space size, provably without aecting optimality. These line graph simpli-
cation rules can be fully applied in polynomial time and are independent of the actual
method used to nd the optimal ordering. Their attractiveness additionally stems from
the fact that they always yield a set of MLNCM subproblems. These may then be solved
in parallel using an appropriate method, which may be heuristically selected. We also
describe cases where these simplication rules already produce an optimal line ordering
solution, as the input graph decomposes into a set of simple components, each with a
search space size of just 1. Finally, we conduct an extensive experimental evaluation of
our methods on several real-world networks and demonstrate that they are fast enough
to be used interactively, for example in a map editor. Most of the results of this chapter
have been previously published in [19] and [20].
4.1 Preliminaries 105
L(e) = {A, B, C}
e
B
A
C
u
f
A
C
д
B
A
C
v
1
2
3
3
2
1
1
2
1
2
3
L(f ) = {A, C}
L(д) = L(e)
MLNCM
L(e) = {A, B, C}
e
B
C
A
1
2
3
3
2
1
u
f
A
C
L(f ) = {A, C}
L(д) = L(e)
д
B
A
C
v
1
2
B
A
C
1
2
3
MLCM
Figure 4.2: Left: In our MLNCM formulation, we require a single ordering p
e
of lines
on an edge e = {u,v}. This ordering gives the position of a line l ∈ L(e) on e. This
position may be stated either w.r.t. u (then denoted as p
u
e
) or v (then denoted as p
v
e
).
Crossings occur at nodes (here, A and C cross at u). Right: In contrast, the original MLCM
formulation allows for two orderings per edge e, which are not necessarily compatible
(if they are not, a crossing occurs on e). Additionally, line orderings of adjacent edges
must be compatible. For example, the crossing between A and C may not occur between
edges f and e at node u, but has to occur on edge e.
4.1 Preliminaries
Denition 4.1 (Line Ordering Solution). Given an edge e = {u, v} ∈ E labeled with
lines l ∈ L(e), we dene p
u
e
(l) = 1, . . . , |L (e)| to give the position of l on e, in clockwise
fashion with respect to u. Each 1, . . . , |L(e)| is called a port of u for e. If the function
p
u
e
: L(e) 7→ {1, . . . , |L(e)|} is bijective (if it is a permutation of L(e)), we say p
u
e
is a line
ordering of e. Correspondingly, p
v
e
(l) is the clockwise position of l on e w.r.t. v. We
require ∀l ∈ L(e) : p
u
e
(l) = |L(e)| − p
v
e
(l) + 1. For notational convenience we dene the
following: p
e
(l) = {p
u
e
(l), p
v
e
(l)},
←−
p
u
e
(l) = p
v
e
(l), and
←−
p
v
e
(l) = p
u
e
(l). We denote the set of all
line orderings for a given line graph G by σ and call it a line ordering solution.
Figure 4.2, left, gives an example.
4.1.1 Crossing Types
We rst lay out the types of crossings that may occur between lines in a metro map. If
we assume the graph to be planar, there are exactly 3 types.
1. Intra-Path Crossings Let A, B be two lines on edge e. If both A and B extend over
a node v to an adjacent edge e
0
(if they both follow the same local path), a crossing
between A and B will occur if
←−
p
v
e
(A) <
←−
p
v
e
(B), but p
v
e
0
(A) < p
v
e
0
(B), or vice versa. See
Figure 4.3.1 for an example.
106 Chapter 4. Line Ordering Optimization
{A, B, C}
{A, B}
{D}
{C, D}
v
3
2
1
1
2
1
2
1
e
e
0
(1)
{A, B, C}
{B, D}
{A, C }
v
1
2
e
0
e
e
00
3
2
1
1
2
(2)
1
{A, C, D}
{B, D}
{A, C }
v
1
2
f
0
e
e
0
3
2
1
1
2
{B}
f
(3)
Figure 4.3: Three types of line crossings that may occur in a transit map. (1) If two lines
A and B on an edge e extend over a node v into an edge e
0
, a crossing appears if their
relative orderings dier. (2) If two such lines extend into two edges e
0
and e
00
, whether
a crossing appears only depends on the ordering of A and B on e. (3) If A and B locally
follow crossing paths, a crossing is unavoidable.
2. Split Crossings Let A, B be again two lines on edge e. If A and B extend over a
node v to two edges e
0
and e
00
(Figure 4.3.2), a crossing occurs if the ordering on
e does not match the ordering induced by the circular edge ordering of e
0
and e
00
.
Let π
v
e
(f ) be the position of f in the circular edge ordering around v, starting in
clockwise fashion at e (that is, π
v
e
(e) = 1). If A extends to e
00
, and B to e
0
, a crossing
will occur if
←−
p
v
e
(A) >
←−
p
v
e
(B), but π
v
e
(e
0
) < π
v
e
(e
00
), or vice versa. For example, in
Figure 4.3.2, a crossing will occur if
←−
p
v
e
(A) >
←−
p
v
e
(B).
3. Ordering-Independent Crossings Let A be a line extending from an edge e to
an edge e
0
, and B a line extending from an edge f to an edge f
0
(Figure 4.3.3).
Additionally, let π
v
e
(f ) > π
v
e
(e
0
), and π
v
e
(f
0
) < π
v
e
(e
0
) (paths (e, e
0
) and (f , f
0
) cross
each other at v). A crossing between A and B is induced by the network topology
and independent of the actual line ordering.
In the context of this work, we ignore ordering-independent crossings. In particular,
this means that we will never count them as crossings.
4.1.2 Metro Line Crossing Minimization
A dierent formulation of the line crossing minimization problem, called the Metro Line
Crossing Minimization (MLCM) was rst dened by Benkert et al. in [26]. Instead of
line crossings occurring at nodes (as is the case in MLNCM), they occur on edges:
Denition 4.2 (Metro Line Crossing Minimization (MLCM)). Given a line graph G =
(V , E, L, L). For each set of lines L(e), nd (not necessarily compatible) orderings p
v
e
(l)
and p
u
e
(l) for each e ∈ E such that the global number of line crossings on edges is min-
imized. For each edge pair adjacent at a node v, their line orderings at v must be com-
patible (they must not introduce any crossings).
The intuition behind this formulation was that line crossings are obfuscated in the
nal drawing by stations, and should thus be moved to the edges (Figure 4.2, right). In
4.1 Preliminaries 107
MLNCM
𝑣
𝑢
𝑒
𝐴
𝐵
𝐶
MLCM
𝑣
𝑢
𝑒
𝐴
𝐵
𝐶
Figure 4.4: A line graph G with line orderings in our MLNCM formulation (left) and the
MLCM formulation (right). Under MLNCM, we may move the crossing between lines A
and B to node v and “behind” line C ending at v. The optimal ordering has 1 crossing.
In contrast, the MLCM problem does not allow a crossing to occur at v, and we have to
cross on the edge e. This requires an additional crossing between C with either A or B.
contrast, our MLNCM formulation is conceptually simpler and only allows crossings at
nodes. In particular, we do not require the additional constraint of adjacent edges hav-
ing compatible line orderings. When compared to our formulation, MLCM has several
additional drawbacks:
1. In real-world transit maps, lines often cross at non-station nodes where previously
paired lines diverge, for example at large intersections in the physical road or rail
network. Placing crossings there often looks natural. Such non-station nodes were
not considered in the original formulations of MLCM.
2. Real-word transit maps also often have lines crossing at station nodes.
3. Real-world transit maps almost never depict crossings on edge segments.
4. Having two orderings per edge complicates the map rendering process.
5. It is not clear from the ordering where an eventual crossing between two lines
should appear on the rendered edge.
At rst glance, the two denitions appear similar, with crossings simply moved from
nodes (in MLNCM) to edges (in MLCM). Their subtle dierence is captured by the fol-
lowing lemma:
Lemma 4.1. An input line graph G may allow for an optimal line ordering with n crossings
under the MLNCM formulation, but only for an optimal number m > n of crossings under
the MLCM formulation.
Proof. Consider the example in Figure 4.4. Under the MLNCM formulation, the optimal
number of crossings is 1. The crossing between A and B may “hide” behind line C ending
atv, as it may occur in v. Under the MLCM formulation, the optimal number of crossings
is 2, as the unavoidable crossing between A and B is not allowed to occur in v, but only
on edge e, where a crossing of C with either A or B is unavoidable.
108 Chapter 4. Line Ordering Optimization
𝐴
𝐶
𝐵
𝐷
𝑢
𝑤
𝑣
𝐴
𝐶
𝐵
𝐷
𝑢
𝑤
𝑣
Figure 4.5: Left: A crossing-minimal MLNCM instance. Lines A, B, and D end between
pass-through lines at nodes w and u. Right: The periphery condition is fullled and lines
A, B, and D all end on the outsides of their terminating edge.
4.1.3 Periphery Condition
Previous work often considered the so-called periphery condition for the placement of
terminus lines. Under this condition, lines terminating at a node v are required to be
positioned at the “outsides“ of an edge. This is always guaranteed for edges with |L(e)| ≤
2 and for edges on which all lines end at node v.
For example, in Figure 4.5, left, lines B and D end between lines passing through their
terminus nodesu and w, which might obfuscate the fact that they end there. In Figure 4.5,
right, the periphery condition is enforced and A, B and C end at the outsides of the
respective edges. We do not consider the periphery condition in this work, but introduce
a dierent concept (line separations) in Section 4.4, which in many cases (although not
all) also ensures the periphery condition.
4.2 Related Work
As mentioned above, previous work in metro-map drawing considered the strongly re-
lated Metro Line Crossing Minimization problem (MLCM), where the goal is to nd two
orderings per input edge, one for each side. In this conguration, the problem was intro-
duced by Benkert et al. in [26]. While the authors could not produce a globally optimal
ordering, the subproblem of nding the optimal line ordering along a single edge was
studied. Here, an edge e = {u, v} is given together with the lines L(e) that traverse the
edge. Additionally, the line orderings at u and v (as induced either by the network topol-
ogy, or by already found edge orderings for other edges adjacent to u or v) are given. If
all l ∈ L(e) pass through u and v, the orderings at u and v are already xed, and there
is nothing left to do. However, if some lines have u and/or v as a terminus, the problem
amounts to nding the optimal insertion order of the lines ending at u and v that in turn
induces a crossing-optimal placement of the remaining lines. A O(n
2
)dynamic program-
ming algorithm for nding the optimal orderings at u and v was presented, where n is
the number of lines passing through u and v. Finding a globally optimal ordering was
left as an open problem.
4.2 Related Work 109
The term MLCM was coined by Bekos et al. in [25] and two subproblems were intro-
duced and studied: In the MLCM problem with line terminals at station ends (MLCM un-
der the periphery condition, MLCM-P, sometimes also called MLCM-SE), all lines ending
at a terminus nodev are placed at the top or the bottom of the ordering, and pass-through
lines are placed in-between. In the MLCM problem with terminals at xed station ends
(MLCM-PA, sometimes called MLCM-FixedSE), the information whether a line ends at
the top or the bottom of an edge is given. By reduction from the Fixed Linear Crossing
Number problem, the authors show that already on simple path graphs, MLCM-P is NP-
hard. The authors then give a polynomial time algorithm to solve MLCM-PA on simple
path graphs and trees. Additionally, they present a polynomial time algorithm to solve
MLCM-P on trees if all lines end at nodes of degree 1. General graphs and trees with
terminals at internal nodes were not considered.
In [11], Asquith et al. extended the polynomial time algorithm results for MLCM-PA
of Benkert et al. to general graphs. They also coined the term periphery condition for
the requirement that all lines must end either at the top or bottom of edges. For the case
where line terminals were not xed, an ILP was given.
Argyriou et al. introduced another interesting variant of MLCM in [10]. Here it is re-
quired that all lines terminate at degree-1 nodes (MLCM-T1). They presented an O((|E|+
|L|
2
)|E|) algorithm for the general MLCM-T1 problem, and an O(|V ||E| +
e∈E
(|L(e)|))
algorithm for the additional special case where each station is represented as a rectan-
gle, and lines can only enter to the left or right of a node. This was further improved by
Nöllenburg in [116], who presented an O(|L|
2
· |V |) algorithm for both MLCM-T1 and
MLCM-PA which iteratively inserts new lines into an initially empty drawing.
For several years, it was an open problem whether the general unconstrained MLCM
problem was also NP-hard. Fink and Pupyrev answered this question in 2013 in the
armative and provided a reduction proof from MLCM-P [59]. We will extend this
proof so that it also works for our MLNCM problem, hence proving its NP-hardness.
Fink and Pupyrev also provided a polynomial-time algorithm for deciding whether a
general MLCM instance allows for a crossing-free line ordering, and if so, obtain that
solution.
Fink et al. observed in [60] that crossings between line bundles may be scattered
along one or more edges, without changing the global number of crossings. This may
lead to results which are not esthetically pleasing and hard to read. To solve this issue,
they grouped crossings into so-called block crossings. A block-crossing is a crossing be-
tween neighboring line bundles following the same edges. They showed that the prob-
lem of minimizing block crossings was also NP-hard. As an open problem, they stated
the problem of distributing block crossings optimally and observed that line bundles fol-
lowing a common sub-path should cross as close to the end of their common sub-path
as possible. We will partly replace the concept of block crossings by the more general
concept of avoiding line separations.
110 Chapter 4. Line Ordering Optimization
Table 4.1: Overview of the complexities for MLCM and MLNCM subproblems.
Problem Graph Compl. Proof Description
MLCM
general
path
NP-hard [59] No restrictions, minimize number of line
crossings.linear (trivial)
MLCM-P
general
path
tree
2
NP-hard [25]
Terminus lines must be grouped at the
top or bottom of edges.
NP-hard [25]
polynomial
1
[25]
MLCM-PA
general
linear
1, 2, 3
[116]
Like MLCM-P, but line terminus
positions (either top or bottom) are xed.
MLCM-T1
general
linear
1, 2, 3
[116]
All lines must terminate at degree 1
nodes.
MLNCM
general
path
tree-like
1, 2
2-tree-like
1, 2
NP-hard Thm. 4.3
No restrictions, minimize numb er of line
crossings.
linear (trivial)
linear
3, 4
Cor. 4.51
linear
3, 4
Cor. 4.56
MLNCM-W
general
general
1, 2
path
NP-hard Thm. 4.4
Minimize weighted line crossings.NP-hard Thm. 4.9
linear (trivial)
MLNCM-S
general
tree-like
1, 2
2-tree-like
1, 2
? —
Minimize line crossings and separations.linear
3, 4
Cor. 4.49
linear
3, 4
Cor. 4.54
MLNCM-WS
general
general
1, 2
NP-hard Thm. 4.5 Minimize weighted line crossings and
weighted separations.NP-hard Cor. 4.10
1
Lines follow simple paths.
2
Lines always terminate at degree-1 nodes.
3
The maximum number of lines
M is considered a constant.
4
The maximum node degree D is considered a constant.
Our own MLNCM formulation was introduced in [19], along with two ILPs to solve
it. As mentioned above, our formulation imposes a single line ordering on each edge
and only allows crossings to happen at nodes. Two additional variants were presented:
a weighted version of MLNCM, where line crossings may be weighted per node, and a
version which additionally penalized line separations (see the previous paragraph). Sev-
eral simplication rules were given which may be applied to the line graph prior to nd-
ing the optimal line ordering, while maintaining optimality. This was further extended
by several so-called untangling rules in [20]. For an overview over the subproblems of
MLCM and MLNCM, as well as their complexities, see Table 4.1.
In a recent work, Frank et al. [62] studied the drawing of hypergraphs as metro-maps.
In this context, they introduced a variant of our MLNCM problem, namely the Metro
Line Crossing Node Minimization Problem (MLCNM). Here, the goal is not to minimize
the global number of crossings, but the number of crossing nodes (nodes where one or
more crossings appear). Then, of course, an optimal solution could place all crossings at
a single node, making the resulting map very hard to read. Even in this conguration,
4.3 Weighted MLNCM 111
the problem remains NP-hard; the authors provide an NP-hardness proof via reduction
from Vertex Cover.
As mentioned above, problem formulations that are similar to our MLNCM formu-
lation previously appeared in the context of wire routing in electronic designs [75] and
edge bundling [123], although they considered only the simple case were all lines end at
nodes of degree 1. For this setting, Pupyrev et al. described an algorithm similar to our
greedy-lookahead algorithm in Section 4.6.4 [123].
The problem is also related to drawing a (multi-)graph with a minimum number of
edge crossings. The major dierences are that (1) in these works, the goal is to nd
a graph embedding with a minimum number of crossings, and (2) crossings of paths
are not typically not considered. In the case of MLCM and MLNCM, the embedding is
xed and crossings between entire edges are not considered (if edges cross in the input
embedding, the embedding can be made planar by adding additional crossing nodes).
4.3 Weighted MLNCM
It is sometimes desirable to move crossings to preferred locations. In our denition
of a line graph, we explicitly allowed non-station nodes, and we would like to prefer
crossings to happen at large non-station nodes, to ensure easy readability. We might also
want to make crossings for a single line more expensive, for example because the line
should be rendered with greater width. We hence also consider the following variation
of MLNCM:
Denition 4.3 (Weighted Metro Line Node Crossing Minimization (MLNCM-W)). Given
a line graph G = (V , E, L, L) and a weight function w : L
2
×V 7→ R
+
0
, with w(l
1
, l
2
,v) =
w(l
2
, l
1
,v) ∀l
1
, l
2
∈ L
2
∀v ∈ V . A crossing of two lines l
1
, l
2
at a node v is weighted by
w(l
1
, l
2
,v). The goal is then to nd a permutation for each set of lines L(e), e ∈ E such
that the sum of weighted line crossings between adjacent edges is minimized.
Obviously, MLNCM is a sub-problem of MLNCM-W, with w(l
1
, l
2
,v) = 1 for each
v ∈ V and each l
1
, l
2
∈ L
2
.
4.4 Minimizing Line Separations
Minimizing line-crossings alone might not always produce esthetically pleasing line or-
derings. We consider Figure 4.6. For each example, the number of line crossings is 4,
which is indeed optimal. The esthetic quality, however, greatly diers. In particular, the
top two examples make it very hard to follow line courses. Additionally, the unavoid-
able crossing between line bundle {A, B} and line bundle {C, D} is depicted more clearly
112 Chapter 4. Line Ordering Optimization
C: 4 S: 4
(1)
B
A
D
C
C: 4 S: 2
(2)
C: 4 S: 2
(3)
C: 4 S: 0
(4)
Figure 4.6: All 4 line ordering examples have 4 crossings (C), which is optimal. However,
(1) and (2) are very hard to read and not visually appealing. While (3) looks better, the
crossing of the two line bundles is best depicted in (4). Counting line separations (S)
exactly captures this intuitive preference.
3
2
1
1
2
1
e
e
0
1
3
B
A
C
v
v
3
2
1
1
1
e
f
1
B
A
д
Figure 4.7: Left: Lines A and B are separated in v by C. Right: The branching of A and
B into two dierent edges f and д does not count as a separation.
in the lower right version. In this section, we formalize the intuitive preference for the
lower right version using a single additional new concept not considered so far in the lit-
erature: line separations. We also describe several additional problems that can be solved
via penalizing line separations. Informally, a line separation occurs at a node v between
edges e and f if lines l
1
and l
2
were placed next to each other on edge e = {u, v}, but not
anymore on edge f = {v, w}. Figure 4.7 gives an example. Formally:
Denition 4.4 (Line Separation). Let e = {u,v}and f = {v, w} be two edges adjacent at
a node v, and let l
1
and l
2
be two distinct lines on e and f , that is l
1
, l
2
∈ L(e), l
1
, l
2
∈ L(f ),
and l
1
, l
2
. If |p
v
e
(l
1
) − p
v
e
(l
2
)| = 1 and |p
v
f
(l
1
) − p
v
f
(l
2
)| , 1 or vice versa, we say lines l
1
and l
2
separate in v.
In Figure 4.6, (1) has 4 line separations, (2) has 4 line separations, (3) has 2 line sep-
arations, and (4) has zero line separations. A ranking based on both the number of line
crossings and line separations therefore exactly matches our intuitive preference, and
(4) is the clear winner.
We now reconsider the periphery condition example from Figure 4.5. The solution
4.4 Minimizing Line Separations 113
𝐴
𝐶
𝐵
𝐷
𝑢
𝑤
𝑣
C: 0 S: 2
𝐴
𝐶
𝐵
𝐷
𝑢
𝑤
𝑣
C: 0 S: 0
Figure 4.8: Left: Crossing-free line ordering, the periphery condition is violated two
times. Right: If line separations are also minimized, the ordering satises the periphery
condition.
𝑢
𝑤
𝑣
C: 0 S: 2
𝑢
𝑤
𝑣
C: 0 S: 0
Figure 4.9: Left: Separation-free line ordering, but the periphery condition is violated.
Right: Extreme example of a crossing and separation free ordering, in which the periph-
ery condition is violated 6 times.
violating the periphery condition has two separations, the solution satisfying the pe-
riphery condition has 0 separations (Figure 4.8). However, avoiding line separations
might not always lead to solutions which satisfy the periphery condition. As a counter-
example, consider Figure 4.9, left. The separation-free solution still violates the periph-
ery condition, and there would be an alternative separation and crossing free line order-
ing which ensure the periphery condition by placing the green line at the bottom. An
extreme case can be seen in Figure 4.9, right.
We now dene the following extension of our original MLNCM problem:
Denition 4.5 (Metro Line Node Crossing and Separation Minimization (MLNCM-S)).
Given a line graph G = (V , E, L, L). For each set of lines L(e), nd a single ordering for
each e ∈ E such that the global number of line crossings and line separations between
adjacent edges is minimized.
Just like for MLNCM, we would like to be able to prefer certain nodes for line sepa-
rations to happen, or to give line separations a higher weight. We hence also consider
the following problem, of which all MLNCM variants presented so far are a subproblem:
Denition 4.6 (Weighted Metro Line Node Crossing and Separation Minimization (ML-
NCM-WS)). Given a line graph G = (V , E, L), a crossing weight function w
×
: L
2
×V 7→
R
+
0
, and a separation weight function w
||
: L
2
×V 7→ R
+
0
. Again we require w
×
(l
1
, l
2
,v) =
w
×
(l
2
, l
1
,v) and w
||
(l
1
, l
2
,v) = w
||
(l
2
, l
1
,v) ∀l
1
, l
2
∈ L
2
∀v ∈ V . For each set of lines L(e),
nd a permutation for each e ∈ E such that the global sum of weighted line crossings
and line separations between adjacent edges is minimized.
114 Chapter 4. Line Ordering Optimization
C: 18 S: 0
Figure 4.10: Optimal line ordering when only line separations are considered. The
resulting ordering is not very readable, many crossings could even be avoided without
adding new line separations.
At this point, a valid question might be whether we need to count crossings at all if
we consider line separations - would it maybe be sucient to only penalize line separa-
tions? Figure 4.10 gives a clear counter-example. Although the depicted line ordering is
completely separation free, it is obviously not optimal, and many crossings could even
be avoided without introducing additional line separations.
4.5 NP-Hardness Results
A direct consequence of Lemma 4.1 is that there is no straightforward way to trans-
late an optimal MLNCM solution to an optimal MLCM solution, impeding a simple NP-
hardness proof for MLNCM by reduction from MLCM. Fortunately, the original NP-
hardness proof of MLCM can be extended to also work for MLNCM and MLNCM-W,
which we lay out in the following.
We now say a line l uniquely extends from an edge e over a node v if l does not
branch at v, and rst prove the following lemma:
Lemma 4.2. Given a line graph G = (V , E, L, L), a line ordering solution σ, and an e dge
e = {u, v}, where a bundle B of lines in L(e) (lines that are positioned next to each other)
uniquely extends over u and v into edges f = {t, u} and д = {v, w}. Figure 4.11 gives an
example. We can group any crossings that occur in u or v b etween lines in B in one of u or
v without creating new crossings.
Proof. We rst observe that by swapping lines in B on e, we might only introduce new
crossings at u or v between lines in B itself. Hence we can ignore other lines. All
unavoidable line crossings that must occur in u and v are induced by the ordering of B
in f and д. If we set the ordering of B on e so that it is compatible with the ordering
of f , only the unavoidable crossings will occur in v, and no crossings occur in u. If
we set the ordering of B on e so that it is compatible with the ordering of д, only the
unavoidable crossings will occur in u, and no crossings occur in v. As the summed
4.5 NP-Hardness Results 115
u
v
u
v
e
f
д
e
f
д
B
n
B
n
Figure 4.11: Line bundle B extends over both u and v. All crossings that occur in u or
v between lines in B (left) can be clustered in u or v (right) without creating any new
crossings.
number of crossings in u and v is now exactly the number of unavoidable crossings
induced by the ordering on f and д, no new crossings could have been added.
Theorem 4.3. The decision variant of MLNCM is NP-hard.
Proof. Similar to the original NP-hardness proof of MLCM, our proof is a reduction
from MLCM-P (the Metro Line Crossing Minimization problem under the periphery
constraint), shown to be NP-hard even on path graphs in [25]. We consider an input
line graph G = (V , E, L, L), which is a path. Figure 4.12, left shows a single edge e of
such a path graph with a crossing-optimal ordering for MLCM under the periphery con-
dition (the ordering of lines extended over u and v depends on the rest of the graph).
To translate this problem into an MLNCM instance, we now build an extended graph
G
0
= (V
0
, E
0
, L
0
, L
0
), in which we rst split each edge e into a left edge e
1
= {u, x
e
}, and
a right edge e
2
= {x
e
,v}, where x
e
is a new node which models crossing on the original
e in our MLNCM formulation. To ensure that lines ending at either u or v are always
placed at the top or bottom, we now employ a technique from [25]: we add nodes u
e
t
and u
e
b
at the top and bottom of u, and nodes v
e
t
and v
e
b
at the top and bottom of v (Fig-
ure 4.12, right). We then add a bundle A of unique lines following path (u
e
t
, x
e
,v
e
b
) and
a similar bundle B of again unique lines following path (u
e
b
, x
e
,v
e
t
), with |A| = |B| = `.
Obviously, A and B have to cross either at u, x
e
or v, adding `
2
unavoidable crossings
per initial input edge e to any crossing-minimal solution. Additionally, each line extend-
ing over both u and v also has to cross A and B, again osetting any crossing-minimal
solution by n
e
· 2`, where n
e
is the number of lines on e that do not end at either u or v.
However, lines ending at u or v may circumvent a crossing with either A or B by ending
at the top or bottom of the respective node (see again (Figure 4.12, right). If a line l ends
at both v and u, it does not have to cross any line in the original graph G, and we can thus
ignore such lines without aecting optimality. If we now choose ` large enough such
that crossing A or B always leads to a non-optimal solution, any solution violating the
periphery constraint is not optimal. However, our MLNCM formulation would allow A
and B to cross, for example, at u instead of x
e
. Then a terminating line could violate the
116 Chapter 4. Line Ordering Optimization
u
v
e
v
u
u
e
tk
u
e
bk
u
e
t
u
e
b
v
e
tk
v
e
bk
v
e
t
v
e
b
e
1
e
2
x
e
A
B
A
u
B
u
A
v
B
v
Figure 4.12: Left: Part of an MLCM-P instance on a path graph G. Lines ending at u
and v are required to end either at the top or bottom of the node. The ordering of pass-
through lines depends on the rest of the graph. Right: An equivalent MLNCM instance
on an extended graph G
0
. We model crossings on edges by an additional node x
e
which
splits edges e into a left edge e
1
and a right edge e
2
. To ensure the periphery condition,
we add four additional nodes u
t
, u
b
, v
t
and v
b
and add suciently large line bundles A
and B in such a way that each pass-through line has to cross them all, but lines ending at
a station may circumvent the crossings if they end at the top or bottom. The additional
keeper nodes u
tk
, u
bk
, v
tk
, v
bk
and keeper line bundles A
u
, B
u
, A
v
, and B
v
are needed
to prevent a crossing between the bundles at nodes u and v.
periphery condition, while still only crossing one of A and B. To prevent this, we fur-
ther extend G
0
and add two keeper nodes u
e
tk
and u
e
tb
to u, and two keeper nodes v
e
tk
and
v
e
tb
to v. For u, they are placed to the left of u
e
t
and u
e
b
, for v, they are placed to the right
of v
e
t
and v
e
b
. We then add keeper bundles A
u
following path (u
e
tk
, u, x
e
), A
v
following
path (v
e
bk
,v, x
e
), B
u
following path (u
e
bk
, u, x
e
) and B
v
following path (v
e
tk
,v, x
e
), all with
` lines. Any crossing of lines extending over u and v with any keeper bundle is unnec-
essary. Additionally, any crossing of keeper bundles with A and B is also unnecessary
- it can be avoided by A and B crossing at x
e
. If ` is large enough, the keeper bundles
therefore enforce that A and B cross at x
e
, and thus ensure the periphery condition.
At each node, each line pair may cross at most once, so this is obviously satised for
` = |V | · |L|
2
. We therefore set ` = |L|
2
. Any avoidable crossing with A, B or the
keeper bundles then adds more crossings to G
0
than could be saved between original
lines in L. Any optimal MLNCM solution of G
0
hence satises the periphery condition.
We note that the lines extending over u and v may now still cross at u or v, which is
not allowed under the MLCM formulation. However, because of the enforced periphery
condition, we may indeed only encounter crossings between such pass-through lines.
If such pass-through lines are present in a node v, they also pass through an adjacent
crossing node x
e
, and therefore Lemma 4.2 applies to v and x
e
. We can thus always move
any crossings on v to x
e
without aecting optimality. Any optimal MLNCM solution for
G
0
can thus be transformed into a feasible and optimal MLCM-P solution for G.
More specically, any crossing-optimal solution for G
0
is osetted by K = |E| · `
2
+
4.6 Heuristic Approaches 117
e∈E
(n
e
+ |L(e)| ·`). If there is an optimal MLNCM solution for G
0
with k + K crossings,
there is hence an optimal MLCM-P solution for G with k crossings, and vice versa.
Theorem 4.4. The decision variant of MLNCM-W is NP-hard.
Proof. Obviously, we can reduce an MLNCM instance to MLNCM-W if we set all weights
to 1.
Theorem 4.5. The decision variant of MLNCM-WS is NP-hard.
Proof. Again, we can reduce any MLNCM instance to MLNCM-WS if we set all weights
for crossings to 1, and for separations to 0.
Theorem 4.6. The decision variants of MLNCM, MLNCM-W, and MLNCM-WS are NP-
complete.
Proof. We can obviously count the crossing and separation score of a given solution in
polynomial time by comparing the positions of all line pairs on all adjacent edges. As
MLNCM, MLNCM-W and MLNCM-WS are NP-hard, they are thus also NP-complete.
Note that an NP-hardness proof for MLNCM-S, that is the special case of MLNCM-
WS where both crossings and separations are always weighted by 1, is missing from this
section. Unfortunately, the NP-hardness proof of MLNCM from above does not work
for MLNCM-S anymore. The dilemma is that if we solve the graph G
0
with MLNCM-S,
it does not seem to be guaranteed that the resulting edge ordering is crossing-minimal
(that is, it might be that we introduced an additional crossing to save a separation). We
consider the complexity of MLNCM-S an interesting open problem.
4.6 Heuristic Approaches
Motivated by the NP-hardness results from above, and as alternatives to the ILP formu-
lation given below, this section describes several baseline heuristic approaches to nd
good solutions for MLNCM-WS (and therefore its subproblems MLNCM, MLNCM-W,
and MLNCM-S). All heuristic approaches will be evaluated in Section 4.11.2. As most of
the heuristic approaches have to repeatedly calculate the target function, we rst quickly
describe how it can be eciently determined for MLNCM, MLNCM-S, and two relevant
special cases of MLNCM-W and MLNCM-WS.
118 Chapter 4. Line Ordering Optimization
v
A
B
C
D
A
B
C
D
e
f
p
v
e
(l
1
) = 2, p
v
e
(l
2
) = 1, p
v
e
(l
3
) = 4, p
v
e
(l
4
) = 3
m = (2, 1, 4, 3)
S
C
C
u
w
v
e
f
A
B
C
A
B
C
D
p
v
e
(l
1
) = 2, p
v
e
(l
2
) = 3, p
v
e
(l
3
) = ∅, p
v
e
(l
4
) = 1
m = (2, 3, ∅, 1)
C
C
u
w
Figure 4.13: Counting the number of crossings between the lines on two edges e
and f is essentially equivalent to counting the number of inversions of a list m =
(p
v
e
(l
1
), ..., p
v
e
(l
|L(f )|
)) where l
i
is the line on f at position i w.r.t. w, and p
v
e
(l) is the po-
sition of line l on e w.r.t v. The number of separations introduced at e (not at f ) is the
number of pairs i and i + 1, with |m[i] −m[i + 1]| > 1. Top: The straightforward variant
where L(e) = L(f ). Bottom: If L(e) , L(f ), we have to insert placeholders into m to
not distort the number of separations (note that no separation between lines A and B is
induced at v in e, as lines A and B were already separated in f by D).
4.6.1 Ecient Calculation of the Target Function
We rst consider the case of calculating the score of an MLNCM-S instance at a single
degree-2 node v adjacent to edges e and f s with L(e) = L(f ) (Figure 4.13). The naive
way to count the number of crossings and separations at v would be to take all O(|L(e)|
2
)
pairs of dierent lines and compare their position in e to their position in f .
A more sophisticated approach is to take e as a reference, and consider the ordering
of f with respect to the ordering of e. Let p
v
e
(l) be the position of line l on e w.r.t. v, and
let l
i
be the line at position i on f w.r.t. w. Then the number of crossings at v is simply
the number of inversions (that is, the number of pairs which are not in order) in the list
m = (p
v
e
(l
1
), p
v
e
(l
2
), . . . , p
v
e
(l
|L(f )|
)).
Given some list a of n elements on which a total order exists, the number of inver-
sions inv(a) in a can be calculated in O(n log n) with a small extension to the merge sort
algorithm [96]: if we merge two (ordered) lists a
l
(the left list) and a
r
(the right list) using
the standard linear time algorithm, each time we encounter an element in a
l
at position
i which is greater than the current element in a
r
at position j, we know that all elements
to the right of a
l
[i] are also greater than a
r
[j], and therefore we have |a
r
| −i inversions.
Summing these local inversion numbers over all merge steps gives the total number of
inversions in quasilinear time.
The number of separations caused at v in e is simply the number of adjacent elements
in m for which |m[i]−m[i + 1]| > 1 and can be determined in linear time. For the number
of separations caused at v in f , we use the same approach, but now with f as a reference.
4.6 Heuristic Approaches 119
This approach can be extended to also work on adjacent edges e and f if L(e) , L(f ):
if no p
v
e
(l) exists because line l is only present in f , but not in e, we simply ignore it.
However, this approach would now introduce erroneous separations as two lines which
were not next to each other in e might now appear next to each other in m at positions
i and i + 1, with |m[i] −m[i + 1]| > 1. To solve this, we build a second list m
0
in which
we insert a placeholder value for each line l < L(e) and only consider adjacent line pairs
in which none of the elements is a placeholder (see Figure 4.13 for an example).
It is easy to see that this approach can be used to count all intra-path crossings and
separations at v by simply applying the method above to all O(deg(v)
2
)pairs of dierent
edges. To also count split crossings in which two lines branch at v for a reference edge
e, we build a list m
∗
similar to the list m described above, but now based on the adjacent
edges in clockwise order, and again count the number of inversions. In particular, note
that we also count intra-path crossings this way.
To nally get distinct values for the number of split crossings, intra-path crossings,
and separations, we simply subtract the number of intra-path crossings determined from
the individual m from inv(m
∗
).
For a single node v, we can therefore calculate distinct values for the number of
split crossings, intra-path crossings, and separations, in O(D
2
M log M), where D is the
maximum node degree, and M is the maximum number of lines, an improvement from
the naive O(D
2
M
2
) method described above.
This approach does not work anymore if arbitrary crossing and separation weights
are present for line pairs. If, however, crossing and separation weights are uniform per
node and crossing type, we can calculate the target function like described above. An
additional observation will prove to be useful for the local search approaches described
below: if we adjust the line ordering at some node e = {u, v}, this only aects the number
of orderings or separations introduced at u and v. To compare the target function value
before and after a local change at an edge e, it is therefore enough to evaluate the target
function only at v and u.
4.6.2 Exhaustive Search
Surely the most naive approach to nd an optimal line ordering is to do an exhaus-
tive search. For each edge e, there are |L(e)|! line permutations. An exhaustive search
enumerates all
e∈E
|L(e)|! possible line ordering solutions and keeps track of the best
solution found so far (possibly with an early stop if the minimum score of 0 is encoun-
tered).
120 Chapter 4. Line Ordering Optimization
4.6.3 Greedy Search
Given a line graph G, dene an initial ordering on the edges E. Iterate over E in this
order. For an edge e = {u, v} ∈ E, choose a reference node (either u or v). W.l.o.g. let u
by the reference node. At u, a strict weak ordering < on the lines L(e) is induced by the
adjacent edges not equal to e. For two lines l
1
, l
2
∈ L(e), there are three possibilities: (1)
l
1
and l
2
branch at u into two edges f and д, then < depends on the circular ordering
of f and д at u. (2) l
1
and l
2
follow the same edge f (and no other), then < depends on
the ordering of L(f ) (if settled), or they are incomparable otherwise. (3) l
1
and l
2
do not
extend together over u, or one or both of them branch into multiple edges, then we mark
them as incomparable. If two lines l
1
, l
2
∈ L(e) are incomparable at u, but comparable at
v, base l
1
< l
2
on v. Otherwise, base l
1
< l
2
on some xed global ordering (for example,
based on the lexicographic ordering of the line names). Sort L(e) according to < and
continue with the next edge.
4.6.4 Greedy Search with Lo okahead
An obvious extension of the Greedy Search approach is the following: If in case (2) the
ordering of L(f ), f = {u, w} is not settled, check whether an ordering of l
1
and l
2
is
induced at w, and so on, following the path of l
1
and l
2
until they branch, or until an
edge with an already xed ordering is found. Algorithms following a similar idea have
been described in [75] and [123].
Lemma 4.7. Given a MLNCM instance of a line graph G in which each l ∈ L follows a
simple path, and where each line terminates at a node of degree 1. The greedy search with
lookahead algorithm will nd a crossing-minimal line ordering solution, regardless of the
initial ordering of E.
Proof. Assume no line ordering has been settled yet, and we begin with an edge e =
{u,v}. Let u be the reference node. If we follow a line pair l
1
, l
2
∈ L(e) over u, then
there are exactly two possibilities: either l
1
and l
2
will branch at some point, inducing
an ordering, or they will terminate together at a node t. If they terminate together at t,
they are incomparable, and we can check whether the other node v induces an ordering
of them. If not, they also terminate on the v-side, and thus follow the exact same path
through G. Their relative ordering is therefore irrelevant, and we can treat them as a
single line.
This induced total ordering of L(e) w.r.t. u is optimal: on the u-side, it allows for or-
derings without any crossings between lines in L(e). On the v-side, any induced crossing
is caused by two lines l
1
and l
2
branching into two edges. But as the ordering of l
1
and
l
2
was already induced by a branch on the u-side, any such crossing is unavoidable. For
each subsequently processed edge f , if we follow a line pair l
1
, l
2
∈ L(f ), a third possi-
4.6 Heuristic Approaches 121
u
v
e
v
u
u
e
tk
u
e
bk
u
e
t
u
e
b
v
e
tk
v
e
bk
v
e
t
v
e
b
e
1
e
2
x
e
A
B
A
u
B
u
A
v
B
v
t
u
t
v
Figure 4.14: Illustration of the proof for Theorem 4.9.
bility appears: we arrive at an edge e with an already settled (optimal) line ordering, and
the same argument holds.
In the worst case, this algorithm has the check for each edge |L|
2
line pairs. As we
follow each line pair for each edge through the graph on the search for a branch point
or a settled edge, it looks like our algorithm is also quadratic in the number of edges.
Note, however, that after a search for a branch point or a settled edge for two lines l
1
and l
2
, starting at an edge e and ending at an edge f , we can already store the relative
ordering of l
1
and l
2
for each edge in between. For each line pair, we therefore only have
to process an edge a constant number of times, resulting in a worst-case complexity of
our greedy branch-based algorithm of O(|E| · |L|
2
· D) (D is the maximum node degree
and is needed as we have to check D adjacent edges for a branch in the worst case).
Corollary 4.8. Given a MLNCM instance of a line graph G in which each l ∈ L follows
a simple path, and where each line terminates at a node of degree 1. The Greedy Branch-
Based algorithm will nd a crossing-minimal line ordering solution, regardless of the initial
ordering of E, in linear time, if |L| and D are considered constants.
This matches previous results in the MLCM setting (MLCM-T1, see Table 4.1).
Weighted Case and Line Separations
To also consider crossing weights in our greedy search with lookahead, we always chose
the ordering of an edge e w.r.t. the adjacent node for which the summed crossing costs
are more expensive. For the special case from Corollary 4.8, the greedy search with
lookahead will no longer produce an optimal solution if line crossings are weighted. In
fact, the following holds:
Theorem 4.9. The decision variant of MLNCM-W remains NP-hard on graphs where all
lines follow simple paths and terminate in nodes of degree 1.
122 Chapter 4. Line Ordering Optimization
Proof. We slightly extend the proof for Theorem 4.3 and add a termination node t
u
for
each terminus node u with deg(u) > 1, connected to u by an edge holding all lines T
terminating at u. We then set the crossing weights of each line l ∈ T with any other line
at u to 0. Figure 4.14 provides an example.
Corollary 4.10. The decision variant of MLNCM-WS remains NP-hard on graphs where
all lines follow simple paths and terminate in nodes of degree 1.
As we only consider the relative ordering of line pairs down the path on both sides,
we also cannot check in general whether an ordering will introduce any line separations
later on (it might depend on the relative ordering of lines not on e which will later join
other edges down the path). Line separations are thus not considered in our greedy
search with lookahead.
4.6.5 Steepest-Ascent Hill Climbing
Let σ ∈ Ω be a line ordering solution, and let N (σ ) be the neighborho od of σ. We recall
from Section 1.3.4 that steepest-ascent hill climbing iteratively chooses the neighbor σ
0
with the best (in our case: minimal) score Θ(σ
0
).
An intuitive neighborhood denition is the following: for a solution σ , N (σ)is the set
of solutions obtainable by changing the ordering of only a single edge. This variant has
been evaluated in [19]. However, even then the neighborhood size depends on the sum
of factorials |L(e)|!, and we have to enumerate all of them to nd the best one. In [19],
we fell back to a stochastic approach and only checked 1,000 random line orderings of an
L(e) if |L(e)| was too large (if |L(e)|! > 3 ·10
5
). In this work, we consider an even simpler
neighborhood: N (σ) is the set of solutions obtainable by swapping a single line pair on
a single edge. Then the neighborhood size is O(|E|M
2
), where again M = max
e∈E
|L(e)|.
4.6.6 Simulated Annealing
We again have a neighborhood N (σ ), the set of solutions obtainable from σ by chang-
ing the ordering of a single line pair on a single edge. We always keep a neighbor σ
0
if its score is better than the current one, that is θ(σ
0
) < θ(σ). To get out of local op-
tima, we additionally add a temperature T which decreases with each iteration. If the
randomly selected neighbor σ
0
is worse or e qual to the current one at iteration i (that is
θ(σ
0
) ≥ θ(σ )), we still select it with probability P(σ , σ
0
,T
i
). Following Kirkpatrick et al.
[94], we set P(σ , σ
0
,T
i
) = exp((−θ (σ
0
)−θ(σ))/T
i
), where T
i
is the current annealing tem-
perature at iteration i. Unfortunately, there are no generally accepted best choices of the
temperature scheduling and/or the initial temperature T
0
. In [19], we chose T
0
= 1, 000
and T
i
= T
0
/i.
4.7 Baseline ILP Formulation 123
4.7 Baseline ILP Formulation
To achieve optimal solutions and to also measure the approximation errors of the heuris-
tics described above, we would like to have an ILP that exactly optimizes a given ML-
NCM, MLNCM-W, MLNCM-S, or MLNCM-WS instance. This section rst provides an
introductory baseline ILP formulation for MLNCM, MLNCM-W, and MLNCM-WS in
which each possible line crossing (or separation) is enumerated. We will improve this
ILP in Section 4.8 by removing the need for this explicit enumeration.
4.7.1 Line Positions
To later retrieve the position p
e
(l) for a line l ∈ L(e) on an edge e, we add |L(e)|
2
line
position variables x
elp
∈ {0, 1} for each edge e ∈ E, where p = 1, . . . , |L(e)| is a position
slot on e. We would like to have x
elp
= 1 when line l is assigned to slot p on e, or x
elp
= 0
if not. For each edge e = {u,v}, we store whether the variables x
elp
are given w.r.t. u, or
w.r.t. v. To enforce that each line l is assigned to exactly one position p, we then add the
following set of constraints:
∀l ∈ L(e) :
|L(e)|
p=1
x
elp
= 1. (4.1)
Equation 4.1 is still fullled if all lines are assigned the same position slot. To avoid such
slot overlls, we additionally guarantee that each position slot is assigned exactly one
line with the following set of constraints:
∀p ∈
{
1, ...,
|
L
(
e
)|}
:
l∈L(e)
x
elp
= 1. (4.2)
4.7.2 Counting Line Crossings
To count crossings in the objective function, we have to consider the two relevant cross-
ing types described in Section 4.1.1: intra-path crossings, and split crossings.
Intra-Path Crossings
In the case of intra-path crossings, two lines A and B on an edge e = {u, v} both ex-
tend over w into an edge f = {v, w}. Then A and B cross at v if p
v
e
(A) < p
v
e
(B) and
p
v
f
(A) < p
v
f
(B), or p
v
e
(A) > p
v
e
(B) and p
v
f
(A) > p
v
f
(B). We would now like to have a binary
decision variable x
e f AB
∈ {0, 1} which is 1 if a crossing is induced by A and B at v, or
0 otherwise. We then add x
e f AB
to the objective function, possibly weighted if we have
124 Chapter 4. Line Ordering Optimization
an MLNCM-W instance. To enforce this assignment, we simply enumerate all possible
crossing positions for A and B. Note that we have to consider whether the position vari-
ables on e and f are given w.r.t. v, or w.r.t. to u or w. If they are given w.r.t. to v for both
e and f , a crossing would for example occur if x
eA1
= 1, x
eB2
= 1, x
f A1
= 1, and x
f B2
= 1,
as this would imply p
v
e
(A) < p
v
e
(B)and p
v
f
(A) < p
v
f
(B). If the positions on either e or f are
not given w.r.t. to v, a crossing would for example occur if x
eA2
= 1, x
eB1
= 1, x
f A1
= 1,
and x
f B2
= 1. For each such crossing-inducing variable assignment, we then add con-
straints like the following for the former example:
x
eAi
+ x
eBj
+ x
f Ai
0
+ x
f Bj
0
−x
e f AB
≤ 3, (4.3)
where i, j and i
0
, j
0
are two position pairs which induce a crossing. If all variables x
eA1
,
x
eB2
, x
f A1
, and x
f B2
are set to 1, the only way to fulll the ≤ 3 constraint is to set x
e f AB
to 1. If only one of the variables is 0, x
e f AB
will be 0 if it appears in the objective function
with a positive coecient. If the coecient is 0 (because crossings have been weighted
with 0), its value is irrelevant.
Split Crossings
If A extends overv into an edge f , and B extends into an edgeд, their relative positions on
one side of v are xed by the circular edge ordering of f and д, and their actual position
on f or д is irrelevant. Whether a crossing occurs only depends on their ordering in e.
For such cases, we introduce a binary decision variable x
e f дAB
∈ {0, 1} which should
be 1 if the positions of A and B on e induce a crossing at v. Similar to Equation 4.3, we
then enumerate all position pairs (i, j) of A and B on e which induce a crossing and add
a constraint of the following form for each of them:
x
eAi
+ x
eBj
−x
e f дAB
≤ 1. (4.4)
Like before, if the rst two variables are 1 and the crossing inducing assignment was
chosen, the split crossing decision variable x
e f дAB
has to be 0 to fulll the constraint.
4.7.3 Counting Line Separations
To consider line separations, we introduce a binary decision variable x
e f A||B
which should
be 1 if lines A and B separate when extending from e to f . Again, we enumerate all po-
sition pairs (i, j, i
0
, j
0
) of A and B on e and f which induce such a separation. We then
enforce the desired assignment by adding the following constraint per position pair:
x
eAi
+ x
eBj
+ x
f Ai
0
+ x
f Bj
0
−x
e f A||B
≤ 3. (4.5)
4.8 Improved ILP Formulation 125
4.7.4 ILP Size
For each edge, we need exactly |L(e)|
2
position variables, so O(|E|M
2
) in total. Each
edge also needs exactly |L(e)| constraints of the form given in Equation 4.1, and |L(e)|
constraints of the form given in Equation 4.2, so O(|E|M)in total. There are
|L(e)|
2
unique
line pairs per edge, so we have to consider O(M
2
) line pairs for avoiding crossings per
edge. If we don’t require lines to follow simple paths, A and B might extend together
into multiple edges, and might therefore cross multiple times in a single node. In total,
we therefore need O(|E|DM
2
) crossing decision variables (both crossings and intra-path
crossings are considered here), where D is the maximum node degree of the input line
graph.
For Equations 4.3, 4.4, and 4.5, we have to consider all possible position pairs for two
lines A and B on edges, so never more than
M
2
< M
2
per line pair and edge. For split
crossings, we indeed only require at most
M
2
constraints of the form given in Equa-
tion 4.3. For intra-path crossings and line separations, we have to consider all position
pairs on both edges. Again, A and B might extend into multiple edges, so we need at most
M
2
D
< M
2D
per line pair, where D is the maximum input node degree. We hence need to
add O(|E|M
2(1+D)
) constraints to count crossings and separations. The total number of
variables is therefore O(|E|DM
2
), and the total number of constraints is O(|E|M
2(1+D)
).
4.8 Improved ILP Formulation
The exponential number of constraints in the baseline ILP formulation is due to explicitly
enumerating all possible position assignments which induce a crossing, for each line pair.
We would like to have a polynomial ILP size. Recall that we perform this enumeration
to test whether p
v
e
(A) < p
v
e
(B) and p
v
f
(A) < p
v
f
(B). If we could eciently check whether
p
v
e
(A) < p
v
e
(B), this would greatly reduce the number of constraints in our ILP.
4.8.1 Range Line Positions
Instead of using a binary decision variable x
elp
which is 1 if line l is positioned at p on
edge e, we introduce a variable x
el ≤p
∈ {0, 1} to decide whether the position of line l
is smaller than p on e. For a single line A positioned at P, we then have a sequence of
such variables for each position slot p = 1, . . . , |L(e)|, with x
el ≤p
= 1 for all p ≤ P, and
x
el ≤p
= 0 for all p > P. To ensure that a position is assigned to l, this sequence has to
eventually arrive at 1, and only switch from 0 to 1 at most once. We enforce the latter
with the following set of constraints:
∀l ∈ L(e) ∀p ∈ {1, ...,
|
L
(
e
)|
− 1} : x
el ≤p
≤ x
el ≤(p+1)
. (4.6)
126 Chapter 4. Line Ordering Optimization
x
e A ≤1
p = 1
e
p = 2
p = 3
p = 4
x
e A ≤2
x
e A ≤3
x
e A ≤4
x
e B ≤2
x
e B ≤3
x
eC ≤3
x
e B ≤4
x
eC ≤4
x
e D ≤4
= 1
Í
= 2
= 3
= 4
1
2
3
4
A
B
C
D
Figure 4.15: An edge e with 4 lines A, B, C, D assigned to positions 1, 2, 3, 4. The ta-
ble shows the non-zero range line position variables in an assignment which satises
Equations 4.6 and 4.6, thus ensuring a bijective mapping from positions to lines.
To ensure that each l is assigned a unique position, we also add the following constraints:
∀p ∈ {1, ...,
|
L
(
e
)|
} :
l∈L(e)
x
el ≤p
= p. (4.7)
So for each position slot p, the number of lines that claim to be positioned before or at
p is exactly p. The constraint for case p = 1 therefore ensures that exactly one line is
assigned to slot 1, the constraint for case p = 2 ensures that an additional dierent line
is positioned at p = 2 (it cannot be the line from position 1 again, because then only one
x
el ≤2
would be 1), and so on. See Figure 4.15 for an example. As each position must thus
be lled with a unique line, all lines get a position, and no position slot is overlled.
4.8.2 Crossing Oracle
To avoid the enumeration of all crossing-inducing position pairs, we would now like to
have variables x
eA<B
, x
eB<A
∈ {1, 0} for a line pair (A, B)telling is whether the position of
line A is smaller than the position of line B on an edge e, or vice versa. This would enable
us to directly check whether a crossing is induced by comparing the respective variables.
Observe that the number of range position variables which are set to 1 directly indicates
how small a position is - if A is positioned at p = 1, then all range position variables
x
eA≤p
will be 1 (see Figure 4.15, right). We can therefore ensure such an assignment with
the following set of constraints:
|
L
(
e
)|
p=1
x
eA≤p
−
p
x
eB≤p
+ x
eB<A
M ≥ 0, (4.8)
x
eB<A
+ x
eA<B
= 1. (4.9)
4.8 Improved ILP Formulation 127
The sum dierence
|
L
(
e
)|
p=1
x
eA≤p
−
p
x
eB≤p
is exactly the (signed) position gap between
A and B. If B is positioned before A, the dierence of the sums is negative (it may never
be smaller than −M), and the only way to fulll the constraint is to set x
eB<A
M to 1. If
on the other hand A is positioned before B, then the x
eA≤p
sum is higher than the x
eB≤p
sum, the dierence is guaranteed to be positive, and x
eB≤B
may be set to zero. To enforce
that it is actually set to zero, Equation 4.9 makes sure that for a line pair (A, B), one of
x
eB<A
and x
eA<B
is set to 1, and the other to zero.
Intra-Path Crossings
Our crossing oracle now enables us to set the intra-path crossing variable x
e f AB
fairly
directly (assuming that the position in e = {u, v} and f = {v, w} are again both given
w.r.t. v):
x
eA<B
−x
f B<A
−x
e f AB
≤ 0 (4.10)
If the relative ordering of A and B is dierent in e than in f , then the absolute dierence
|x
eA<B
−x
f B<A
| is 0, and x
e f AB
may remain 0, as indeed no crossing occurs. If their relative
ordering w.r.t. v is the same, a crossing occurs, and the absolute value dierence is 1,
enforcing us to set x
e f AB
to 1. To model the absolute value computation in our ILP, we
use the following standard replacement:
x
eA<B
−x
f B<A
−x
e f AB
≤ 0, (4.11)
−x
eA<B
+ x
f B<A
−x
e f AB
≤ 0. (4.12)
If x
eA<B
= x
f B<A
, then x
e f AB
may still be zero as before. If x
eA<B
= 1 and x
f B<A
= 0, then
Equation 4.11 will enforce x
e f AB
= 1. If x
eA<B
= 0 and x
f B<A
= 1, then Equation 4.12 will
enforce x
e f AB
= 1.
Split Crossings
For split crossings where A and B extend into dierent edges f and д, we again have
to only consider the relative ordering of A and B at e. Then a crossing either occurs
if the position of A is smaller than B, or vice versa. Hence, we add the following set of
constraints to get the correct assignment for the split crossing variable x
eAB
. If a crossing
would occur if A is positioned before B, we add the following constraint:
x
eA<B
−x
eAB
≤ 0. (4.13)
If a crossing would occur if A is positioned after B, we add the following constraint:
x
eB<A
−x
eAB
≤ 0. (4.14)
128 Chapter 4. Line Ordering Optimization
4.8.3 Line Separations
Just as for our baseline ILP, we would like to have a binary decision variable x
eAkB
∈
{0, 1} which is 0 if
p
v
e
(A) −p
v
e
(B)
= 1 (if A and B are placed next to each other) and 1
otherwise, but without enumerating all separation-inducing position pairs like we did
in Section 4.7.3. A separation may only occur if A and B extend from an edge e into an
edge f . Obviously, if |L(e)| = |L(f )| = 2, A and B can never separate, and we do not
need explicit separation variables. We could then get the desired assignment with the
following set of constraints for each line pair {A, B} on an edge e:
|
L
(
e
)|
p=1
x
eA≤p
−
p
x
eB≤p
−x
eAkB
M ≤ 1. (4.15)
Recall that
|
L
(
e
)|
p=1
x
eA≤p
−
p
x
eB≤p
is the signed position gap of A and B, the absolute
value of which is 1 exactly then if they are placed next to each other. Again, we use the
following replacement for the absolute value computation:
|
L
(
e
)|
p=1
x
eA≤p
−
p
x
eB≤p
−x
eAkB
M ≤ 1, (4.16)
|
L
(
e
)|
p=1
x
eB≤p
−
p
x
eA≤p
−x
eAkB
M ≤ 1. (4.17)
Note, however, that x
eAkB
might always be set to 1. To ensure that it is indeed only set to
1 if A and B are not placed next to each other, we additionally add the following single
constraint per edge e:
(l,l
0
)∈U (e)
x
el kl
0
≤
|
L
(
e
)|
2
−
|
L
(
e
)|
− 1. (4.18)
There are indeed
|
L
(
e
)|
2
unique line pairs on L(e), and exactly
|
L
(
e
)|
− 1 of them will be
placed next to each other. So exactly
|
L
(
e
)|
2
−
|
L
(
e
)|
− 1 lines are separated on e, and
this number is already exactly matched by the sum of variables x
eAkB
which are forced
to have value 1.
Like for crossings, we now want a binary decision variable x
e f AkB
∈ {0, 1} which
should be 1 if A and B separate while extending from e to f , or 0 otherwise. Again, the
desired assignment would be satised with an absolute value constraint of the form
|x
eAkB
−x
f AkB
| −x
e f AkB
≤ 0, (4.19)
4.9 Line Graph Simplication 129
which we again model as follows:
x
eAkB
−x
f AkB
−x
e f AkB
≤ 0, (4.20)
−x
eAkB
+ x
f AkB
−x
e f AkB
≤ 0. (4.21)
Note that if A and B extend from e to f , but |L (e)| = 2, the following constraint is enough:
x
f AkB
−x
e f AkB
= 0. (4.22)
The decision variable x
e f AkB
is then added to the objective function with its weight
reecting the separation penalty between A and B on node v connecting edges e and f .
4.8.4 Improved ILP Size
For each line on each edge, we have |L(e)| range line position variables, so no more than
|E|M
2
in total.
For the crossing oracle, we add O(|E|M
2
) variables of the form x
eB<A
, and asymptoti-
cally just as many constraints (Equations 4.8 and 4.9) to ensure their correct assignment.
We then again add O(|E|DM
2
) split- and intra-path crossing variables, and asymptoti-
cally just as many constraints again (Equations 4.12, 4.11, 4.11, 4.13, and 4.14).
For the line separations, we again add at most |E|M
2
variables of the type x
eA||B
and
just as many constraints to ensure their desired assignments. Finally, we add O(|E|DM
2
)
line separations variables of the type x
e f A||B
, and again just as many constraints for them.
In total, we now have O(|E|DM
2
) variables and O(|E|DM
2
) constraints. As desired, the
ILP size is polynomial now.
4.9 Line Graph Simplication
Real-world input line graphs typically contain segments or paths for which at least a
partial optimal ordering of the lines is intuitively obvious (see Figure 4.41, top, for an
example). This section aims to formalize this intuition into provable lemmas. We then
derive several simplication rules that may be applied to the input line graph G in poly-
nomial time prior to nding optimal line orderings. These simplications rules do not
aect the global line ordering optimality. Generally, this means that we can easily con-
struct an optimal line ordering solution σ on G from an optimal line ordering solution
σ
0
on the simplied graph G
0
. The simplication rules described in this section have the
potential to greatly reduce the overall search space size. Additionally, they may break up
the input line graphs into multiple connected components, which may then be solved by
appropriate methods (either the ILP described above, a heuristic, or even an exhaustive
search), possibly in parallel.
130 Chapter 4. Line Ordering Optimization
A
B
Figure 4.16: If two lines A and B strictly follow the same simple path and have the same
crossing and separation weights at each node, an optimal placement will always be to
put them next to each other, crossing-free, in each edge. This might not be the only
optimal placement, however.
This Section is organized as follows: We rst prove Lemmas 4.11 - 4.15. We then
use them to derive a set of pruning (Section 4.9.2), cutting (Section 4.9.3) and untangling
rules (Section 4.9.4). For each rule, we prove that the transformation does not aect
global optimality. We will then discuss the complexity of a full graph simplication in
Section 4.9.6, and describe several instances for which the simplication rules completely
solve the corresponding MLNCM problem (Section 4.10).
4.9.1 General Observations
We consider Figure 4.16. It is not hard to see that if two lines follow the same simple path
through G, an optimal ordering is to always place them next to each other. However, an
optimal line ordering solution will not necessarily place them this way, see Figure 4.9,
right for a crossing- and separation-free counter-example.
Lemma 4.11. Given a set of lines B = {A, B, C, . . . } ⊆ L which all strictly follow the
same simple path P (lines are not allowed to branch). Additionally, we require crossing and
separation weights for all l ∈ B to be equal in all path nodes. For any ordering l
1
, . . . , l
|B|
on the B a globally optimal line ordering solution exists in which for any two adjacent path
edges e = {u, v}, f = {v, w} and j > 1 it holds that p
v
e
(l
j
) = p
w
f
(l
j
) = p
v
e
(l
j−1
)+ 1 (all l ∈ B
are always positioned next to each other with the same ordering, free of internal crossings
or separations).
Proof. Let σ be a line ordering solution for which the property does not hold, and let
m ∈ B be the line in B that induces the lowest cost (sum of weighted line separations
and crossings). We claim that we may place any l ∈ B \ {m} next to m on the path
at lower or equal cost. Obviously the new position will not introduce any crossings
or separations between l and m. All separations that may be induced by l at the new
position were already induced by m. The weighted crossings l will induce at the new
position will be equal to the weighted crossings m induces. As the total cost induced by
m was smaller than the total cost induced by l, the new solution σ
0
will hence always be
4.9 Line Graph Simplication 131
𝑢
𝑣
𝑤
𝑒
𝑓
𝐴
𝐵
𝐶𝐶
𝑢
𝑣
𝑤
𝑒
𝑓
𝑢
𝑣
𝑤
𝑒
𝑓
𝐴
𝐵
𝐶
𝑢
𝑣
𝑤
𝑒
𝑓
Figure 4.17: Top: If all lines from e distinctly extend over u, and if all crossing and
separation weights at u are cheaper than or equal to those at v, we can base the line
ordering of e on that of f , moving each crossing and separation from u to v. Bottom:
If lines do not distinctly extend over u, moving the crossings to u might now result in
multiplied crossing costs for lines which branch at u.
better than or equal to σ . We can thus produce any permutation of B with all lines next
to each other, and it will be an optimal placement of the B.
The following lemma is similar to Lemma 4.2, but also considers crossing and sepa-
ration weights:
Lemma 4.12. Given a line graph G = (V , E, L, L), a line ordering solution σ , and edges
e = {u, v} and f = {v, w}, with L(e) = L(f ) and deg(v) = 2. If all lines L(e) either end
or uniquely extend over u, and if in the latter case crossing and separation weights between
all lines in L(e) are equal or lower at u than at v, we can make v crossing and separation
free without aecting optimality.
Proof. If we set p
v
e
= p
w
f
, v will be crossing and separation free, and all crossings or
separations that previously occurred at v now occur at u (some may not occur at all if
a line partner has a terminus at u, in which case σ was not optimal). As all crossing
and separation weights are equal to or cheaper to those at u, the thus obtained ordering
solution σ
0
will be equal or better than σ.
Observe that if a line does not distinctly extend over u, crossing costs might be mul-
tiplied there, which is why we excluded such cases. Figure 4.17 illustrates this.
Lemma 4.13. Given an edge e = {u, v} where each l ∈ L(e) has terminates at u. The
ordering of L(e) will not aect the number of crossings or separations in u.
Proof. No l ∈ L(e) extend over v, so they cannot induce any crossing or separation
(Figure 4.18).
Lemma 4.14. Given an edge e = {u, v} with |L(e)| = 1. The ordering of L(e)will not aect
the global number of crossings or separations.
Proof. All orderings of L(e) are equivalent, as there is only one.
132 Chapter 4. Line Ordering Optimization
{A, B, C}
{E, F }
{D, E, F }
{D}
v
1
2
1
e
0
e
3
2
1
1
2
Figure 4.18: Illustration of Lemma 4.13.
{A, B, C}
{A, C}
{D}
{B, D}
v
3
2
1
1
2
1
2
1
e
f
0
f
1
Figure 4.19: Illustration of Lemma 4.15.
We now consider split crossings as dened in Section 4.1.1. The following lemma
directly follows from the denition:
Lemma 4.15. If all lines L (e) for an edge e = {u,v} uniquely extend over v into n edges
f
i
, e, the circular edge ordering of the f
i
induces a partial ordering on the L(e) which does
not induce any crossings or separations at v between two lines l, l
0
∈ L(e) which continue
into dierent edges f
1
and f
2
, namely a partial ordering which ensures
←−
p
v
e
(l) >
←−
p
v
e
(l
0
) if
π
v
e
(f
1
) > π
v
e
(f
2
).
Proof. Per denition, a separation may only occur between lines continuing together
from an edge e into and edge f , so the relative ordering of such line bundles has no
eect on the number of separations at v. As l and l
0
extend into dierent edges, only a
split crossing may occur, but per denition of a split crossing, the partial ordering can
never induce it (Figure 4.19).
4.9.2 Pruning Rules
This section describes several simple rules to prune to line graph G prior to the line
ordering optimization, provably without aecting the optimality of the line orderings
found on the simplied graph G
0
.
In Sections 4.9.3 and 4.9.4, we will describe rules that split line graph nodes. We
therefore rst dene v
∗
as the original input line graph node of v (which may be v
itself). If any of the following rules splits a node v into nodes v
0
and v
00
, the crossing and
separation penalties at v
0
and v
00
will be set to the crossing and separation penalties at
the original node v.
Pruning Rule 1 (Node Contraction). Contract each node v with deg(v) = 2 with adjacent
edges e = {u, v}, f = {v, w } if the following holds:
1. L(e) = L(f )
2. ∀l
1
, l
2
∈ L(e)
2
, l
1
, l
2
: All lines terminate or have lower crossing and separation
weights at u, or all lines terminate or have lower crossing and separation weights at v.
Figure 4.20 provides an example.
4.9 Line Graph Simplication 133
{𝐴}
{𝐵}
{𝐴, 𝐵}
𝑒
𝑢
{𝐴}
{𝐵}
𝑤
𝑣
𝑒
0
{𝐴, 𝐵}
{𝐴}
{𝐵}
𝑢
{𝐴}
{𝐵}
𝑤
𝑒𝑒
0
{𝐴, 𝐵}
Figure 4.20: Illustration of Pruning Rule 1 (Node Contraction). Node v is adjacent to
two edges e and e
0
holding the same lines (L(e) = L(e
0
)). We can contract v if the crossing
and separation costs at u or w are cheaper than or equal to those at v without aecting
optimality.
𝑣
{𝐴, 𝐵, 𝐶}
{𝐷}
{𝐴, 𝐵, 𝐶, 𝐷}
𝑢
{𝐴, 𝐵, 𝐶}
{𝐷}
𝑡
𝑤
𝑣
{𝐾 }
{𝐷}
{𝐾, 𝐷}
𝑢
{𝐾 }
{𝐷}
𝑡
𝑤
𝑒
𝑓
𝑔
𝑒
𝑓
𝑔
Figure 4.21: Illustration of Pruning Rule 2 (Line Partner Collapse). If a group B of lines
(here, B = {A, B, C}) follows the same simple path, we can replace them by a single line
(here: K) without aecting optimality if the crossing weights for the new line are set to
the sum of the crossing weights of each l ∈ B are each aected node.
Lemma 4.16. Pruning Rule 1 (Node Contraction) does not aect the optimality of the line
ordering.
Proof. Lemma 4.12 holds for v. Any optimal solution σ with crossings or separations at
v can thus be transformed into an optimal solution σ
0
where v is crossing and separation
free. We therefore do not have to consider v during optimization.
Pruning Rule 2 (Line Partner Collapse). Replace a set B ⊆ L of lines with a single new
line k if the following holds:
1. All lines l ∈ B follow the exact same simple path P through the line graph.
2. For all l ∈ B and at each path nodes, crossing and separation weights are equal.
Set all crossing weights for k to be the sum of the respective weights for all l ∈ B.
Figure 4.21 provides an example.
Lemma 4.17. Pruning Rule 2 (Line Partner Collapse) does not aect the optimality of the
line ordering.
134 Chapter 4. Line Ordering Optimization
{A, B, C, D, E}
{A, B}
{A, B}
{B}
{A}
v
д
{A, B}
{A, B}
{B}
{A}
u
Figure 4.22: Illustration of Pruning Rule 3 (Single Edge Prune).
Proof. Lemma 4.11 holds, and there will always be an optimal line ordering solution
in which the lines B are placed next to each other with the same ordering in all path
edges. We can thus settle this ordering at random initially, optimize with the placeholder
line k, and insert the initial ordering into the optimal line ordering without aecting
optimality.
Pruning Rule 3 (Single Edge Prune). Delete edges e = {u, v} with deg(u) = deg(v) = 1.
Figure 4.22 provides an example.
Lemma 4.18. Pruning Rule 3 (Single Edge Prune) does not aect the optimality of the line
ordering.
Proof. Per Lemma 4.13, the ordering of L(e) will not aect the number of crossings or
separations in any solution.
4.9.3 Cutting Rules
While the pruning rules may already greatly reduce the search space size of the line or-
dering optimization problem, we would like to partition the input line graph into compo-
nents which can be optimized separately and in parallel. We call a line graph component
whose line orderings does not aect the number of crossings or separations in the rest
of the line graph an ordering-relevant connected component. In this section, we de-
scribe several cutting rules which transform the input line graph in such a way that any
ordering-relevant connected component in the original line graph has a corresponding
connected component in the transformed graph.
Like in Section 4.9.2, we show for each cutting rule that it will not aect the optimal-
ity of the nal line ordering.
Cutting Rule 1 (Single Line Cut). Cut an edge e = {u, v} with deg(u) > 1 and deg(v) > 1
into two edges e
0
= {u, v
0
} and e
00
= {v
00
,v} if
|
L
(
e
)|
= 1. v
0
and v
00
are new nodes
positioned somewhere along the original edge e. Set L(e
0
) = L (e
00
) = L (e).
4.9 Line Graph Simplication 135
{C, D, E}
{D, E}
u
{A, B, C}
{A, B}
v
e
{C}
{C, D, E}
{D, E}
{C}
e
0
u
{A, B, C}
{A, B}
v
e
00
{C}
v
00
v
0
w
t
w
t
Figure 4.23: Illustration of Cutting Rule 1 (Single Line Cut). An edge e with |L (e)| = 1
may be cut as depicted without aecting optimality.
u
{A}
{B}
{A, B}
v
{C}
{C}
w
t
u
{A}
{B}
{A, B}
v
{C}
{C}
w
t
v
0
e
e
0
Figure 4.24: Illustration of Cutting Rule 2 (Terminus Detachment).
Figure 4.23 gives an example.
Lemma 4.19. Cutting Rule 1 (Single Line Cut) does not aect the optimality of the line
ordering.
Proof. Per Lemma 4.13 the ordering of L (e) is irrelevant for the number of crossings or
separations. As deg(v
0
) = deg(v
00
) = 1, no additional crossing or separation may occur
in these new nodes. As L(e
0
) = L (e
00
) = L (e), the number of crossings and separations at
v and u is also not aected.
Cutting Rule 2 (Terminus Detachment). Given a node v adjacent to an edge e = {u,v}
for which the following holds:
1. v is a terminus station for all lines l ∈ L(e).
2. deg(v) > 1.
Detach e from v. (Delete e, add a new node v
0
positioned somewhere along the original edge
e, and add a new edge e
0
= {u, v
0
} with L(e
0
) = L (e)).
Lemma 4.20. Cutting Rule 2 (Terminus Detachment) does not aect the optimality of the
line ordering.
136 Chapter 4. Line Ordering Optimization
f
e
h
i
д
d
a
b
c
{D, E, F }
{A, B, C}
{A, B, C}
{C}
{A, B}
{A, B}
{D}
{F }
{E}
{E}
{I }
{E, F }
{E, F }
G
e
i
a
c
{D, E, F }
{X, C}
{C}
{D}
{F }
{E}
{E, F }
{X }
G
00
h
Figure 4.25: Left: Line graph with 7 lines A, B, C, D, E, F , G. Right: The graph G
0
result-
ing from the repeated application of pruning rules 1-3 and cutting rules 1-2 to G.
v
v
e
д
h
f
{A, A
0
}
{A, A
0
}
{B, B
0
, C, D}
{B, B
0
, C, D}
v
0
e
0
д
0
h
0
f
0
{A, A
0
}
{A, A
0
}
{B, B
0
, C, D}
{B, B
0
, C, D}
v
00
Figure 4.26: Left: Excerpt from the New York subway map. Middle: Corresponding line
graph. Right: Line graph (exact edge geometries have been omitted) after Untangling
Rule 1 was applied.
Proof. Per Lemma 4.13 the ordering of L (e) is irrelevant for the number of crossings or
separations in v. As deg(v
0
) = 1, no additional crossing or separation may occur in
these new nodes. As L(e
0
) = L(e), the number of crossings and separations at u is also
unaected.
4.9.4 Untangling Rules
We now consider Figure 4.26 and 4.27. In Figure 4.26, it is obvious that the line orderings
on h and д can be optimized independently of the line orderings on e and f , but none of
the pruning and cutting rules apply, as the two line bundles {A, A
0
}and {B, B
0
, C, D} cross
each other at v. In Figure 4.27, the situation is slightly more complex, as the crossing of
two line bundles happens over three nodes u, v and w, but it is again intuitively obvious
that the ordering of the line bundles can be optimized independently here. This section
introduces several so-called untangling rules which aim to formalize these intuitions into
formal simplication rules. The name untangling was chosen because in the best case,
these rules “untangle” the line graph until only simple “threads” (with a line-ordering
search space size of 1) remain. Again, each untangling rule is proven to not aect line
ordering optimality.
4.9 Line Graph Simplication 137
w
00
w
u
w
u
v
{A, B, C, F , H }
{A, B, C, D, E, F, G, H }
{A, B, C, D, E, F, G, H }
{D, E, G}
{A, B, C, F , H }
{D, E, G}
e
e
0
i
h
д
f
u
0
{A, B, C, F , H }
{D, E, G}
{A, B, C, F , H }
f
00
i
0
h
0
д
0
f
0
u
00
w
0
{A, B, C, F , H }
f
0
{D, E, G}
{D, E, G}
Figure 4.27: Left: Excerpt from the Stuttgart light rail map. Middle: Corresponding line
graph. Right: Line graph (exact edge geometries have been omitted) after Pruning Rule
1 and Untangling Rule 3 was applied. The number of possible line orderings on e and e
0
was reduced from |L(e)|! · |L(e
0
)|! = 8! ·8! = 1.63 ·10
9
to |L(f
0
)|! · |L(f
00
)|! = 5! ·3! = 720.
v
L
1
= {A, B}
L
1
= {A, B}
L
2
= {C, D}
L
2
= {C, D}
v
0
L
1
= {A, B}
L
1
= {A, B}
L
2
= {C, D}
L
2
= {C, D}
v
e
д
h
f
e
0
д
0
h
0
f
0
Figure 4.28: Illustration of Untangling Rule 1. Left: Two line bundles L
1
and L
2
, with L
1
∩
L
2
= ∅ cross at a node v. The ordering of L
1
on h and д can be optimized independently
of the ordering of L
2
on e and f . Right: Untangling Rule 1 has been applied to v.
Full X Structure
We rst consider situations as depicted in Figure 4.28, left, which we call a Full X struc-
ture, and dene the following untangling rule on them:
Untangling Rule 1 (Full X). Given a node v with deg(v) > 2 and with n > 2 adjacent
edges with at least one line for which v is not a terminus (to remove trivial cases that can
be transformed into cases eligible for Pruning Rule 1 (Node Contraction)).
We consider all pairs {e, f } of adjacent edges e = {v, u} and f = {v, w}. If for such a
pair it holds that L(e) = L (f ), and if all l ∈ L (e) uniquely extend over v, delete e and f , add
a new node v
0
at the same position as v, and add new edges e
0
= {v
0
, u} and f
0
= {v
0
, w }
with L(e
0
) = L (f
0
) = L (e).
Figure 4.28, right gives an example.
Lemma 4.21. Untangling Rule 1 does not aect the optimality of the line ordering.
138 Chapter 4. Line Ordering Optimization
v
L
1
= {A, B}
L
2
= {C}
L
1
∪ L
2
= {A, B, C}
f
д
e
v
0
L
1
= {A, B}
L
2
= {C}
L
1
= {A, B}
f
0
д
0
e
0
v
00
e
00
L
2
= {C}
u
u
0
u
00
Figure 4.29: Illustration of Untangling Rule 2. If a major leg (here edge e) terminates at
node u and branches at node v into two minor legs (here edges f and д), v and e can be
split like depicted.
Proof. No crossing or separation between any line l ∈ L(e) and any line m < L(e) was
possible originally. All crossings or separations between lines l < L(e) may still occur in
v. All crossings or separations between lines l ∈ L(e) that may have occurred in v can
still occur in v
0
.
This rule has two important eects: First, the nodes v and v
0
may now be eligible for
contraction using Pruning Rule 1 (Node Contraction). Second, v and v
0
may now be in
dierent connected components which can be optimized separately.
Single Y Structures
Figure 4.29 depicts a situation in which two distinct line bundles L
1
and L
2
join at a node
v and continue over a single edge e to a node u, which is a terminus for all l ∈ L
1
∪ L
2
.
It is intuitively obvious that any crossing or separation between a line l ∈ L
1
, and a
line m ∈ L
2
at node v can be avoided, as for any line ordering on f and д, there is line
ordering for e which does not induce such a crossing or separation. This intuition is
formalized in the following untangling rule:
Untangling Rule 2 (Full Y). Given a no de v adjacent to an edge e = {u,v} with deg(u) =
1 and the following additional properties:
1. Each l ∈ L (e) terminates at u.
2. Each l ∈ L(e) uniquely extends over v into one of deg(v) − 1 edges e
1
, ..., e
deg(v)
, with
e
i
, e, and all l ∈ L(e
i
)uniquely extend over v into e (that is, L(e
1
)∪...∪L(e
deg(v)−1
) =
L(e) and L(e
1
) ∩ ... ∩L (e
deg(v)−1
) = ∅).
We call e the major leg, and e
1
, ..., e
deg(v)
the minor legs. Split u and v into nodes u
0
, u
00
and
v
0
,v
00
. Connect v
0
and u
0
with an edge e
0
and set L(e
0
) = L(e
1
) (the lines of the leftmost
minor leg). Conne ct v
00
and u
00
with an edge e
00
and set L(e
00
) =
deg(v)−1
i=2
L(e
i
) (the lines
of the remaining minor legs to the right). Connect v
0
to the node that v was originally
4.9 Line Graph Simplication 139
v
{A, B, D}
{C, D, E}
{A, B, C}
f
д
e
{A, B}
e
0
e
00
{C}
u
u
0
u
00
v
{C, D, E}
f
д
{A, B, D}
h
{E}
h
{E}
Figure 4.30: Illustration of Untangling Rule 3.
connected by e
1
with a new edge e
0
1
. Set L(e
1
) = L (e
0
1
). Connect all remaining minor legs in
a similar way to v
00
Note that this rule does not only consider Y structures with two minor legs (as de-
picted in Figure 4.29), but works with any number of minor legs.
Lemma 4.22. Untangling Rule 2 does not aect the optimality of the line ordering.
Proof. Per Lemma 4.15, the minor legs induce a partial ordering on the lines on e which
does not induce any crossing or separation between two lines coming from dierent
minor legs. Splitting v therefore does not prevent any optimal line crossings or separa-
tions at v between lines from dierent minor legs. Per Lemma 4.13, the line ordering on
e has no eect on the number of crossings or separations in u, and as deg(u) = 1, there
cannot be any crossings or separations in u. Splitting u therefore does also not prevent
any optimal line crossings or separations at u. Any crossing or separation at v between
a line on a minor leg, and a line on the major leg is still possible in the corresponding
split node.
To be later able to reconstruct an ordering of the original edge e from the optimal
ordering on edges e
0
and e
00
, we have to store an ordering of e
0
and e
00
with respect
to e. This is just the original clockwise ordering between e
1
and the remainder of the
minor legs. If Untangling Rule 2 (or any of the untangling rules dened below) is applied
repeatedly, we end up with a hierarchy of edge orderings, from which (given optimal line
orderings for the leafs of this tree) the line ordering for the original input graph must be
reconstructed.
A slightly dierent kind of Y-structure, which we call a partial Y, is depicted in Fig-
ure 4.30. Just like in the previous full Y-structure, the lines on the major leg e completely
and uniquely branch into the minor legs f and д. However, the lines on the minor legs
are not pairwise disjoint: a line D is present on f and д. While Lemma 4.15 still holds for
e, we cannot simply split v into distinct nodes per minor legs as before, as separations
140 Chapter 4. Line Ordering Optimization
v
{A, B}
{C}
{A, B, C}
д
f
e
{A, B}
{C}
{A, B}
i
0
h
0
д
0
{C}
u
{A, B}
{C}
h
i
{A, B}
{C}
e
0
e
00
f
0
u
0
u
00
v
00
v
0
Figure 4.31: Illustration of Untangling Rule 4, with inverse minor legs.
or crossings involving D may occur there. We may still split u, however, as captured in
the following rule:
Untangling Rule 3 (Partial Y). Given a node v adjacent to an e dge e = {u, v} with
deg(u) = 1 and the following additional properties:
1. Each l ∈ L (e) terminates at u.
2. Each l ∈ L(e) uniquely extend over v into one of n > 1 edges e
1
, ..., e
n
, e
i
, e, and
each e
i
has a line that uniquely extends into e (that is, L(e
1
) ∪ ... ∪ L(e
n
) ⊇ L(e),
L(e
1
) ∩ ... ∩L (e
n
) ∩ L(e
n
) = ∅ and ∀e
i
: L(e
i
) ∩ L(e) , ∅).
We again call e the major leg, and e
1
, ..., e
n
the minor legs. Split u into nodes u
0
, u
00
. Connect
v and u
0
with an edge e
0
and set L(e
0
) = L (e
1
) (the lines of the leftmost minor leg). Connect
v and u
00
with an edge e
00
and set L(e
00
) =
n−1
i=2
L(e
i
) (the lines of the remaining minor legs
to the right).
Lemma 4.23. Untangling Rule 3 does not aect the optimality of the line ordering.
Proof. Lemma 4.15 and Lemma 4.13 still hold as in the previous proof, we may therefore
split e in the described fashion without aecting optimality.
Note that Untangling Rule 2 only detaches a single minor leg, although the prerequi-
sites allow for an arbitrary number of minor legs. To untangle such structures, a repeated
application of the untangling rules as described above is necessary.
Double Y Structures
We now consider Figure 4.31, left, which is a condensed version of the real-world ex-
ample in the Stuttgart light rail network given in Figure 4.27. We call such structures
Double Y and dene the following untangling rule on them:
Untangling Rule 4 (Full Double Y). Given an edge e = {u, v} with deg(u) = deg(v) ≥ 3
and the following additional properties:
4.9 Line Graph Simplication 141
v
{A, B}
{C}
{A, B, C}
д
f
e
{A, B}
{C}
{A, B}
i
0
h
0
д
0
v
00
{C}
u
u
0
u
00
v
u
{A, B}
{C}
h
i
{A, B}
{C}
e
0
e
00
f
0
v
0
Figure 4.32: Illustration of Untangling Rule 4, with matching minor legs.
1. Each l ∈ L(e)uniquely extends over v into one of deg(v)−1 edges f
1
, ..., f
deg(v)
, f
i
, e,
and all l ∈ L(f
i
) uniquely extend over v into e (that is, L(f
1
)∪... ∪L(f
deg(v)−1
) = L (e)
and L(f
1
) ∩ ... ∩L (f
deg(v)−1
) = ∅).
2. Each l ∈ L (e) uniquely extends over u into one of deg(u)−1 edges д
1
, ..., д
deg(v)
, д
i
, e,
and all l ∈ L(д
i
) uniquely extend over v into e (that is, L(д
1
)∪... ∪L(д
deg(v)−1
) = L (e)
and L(д
1
) ∩ ... ∩L (д
deg(v)−1
) = ∅).
3. There is a bijective mapping A : {f
1
, ..., f
deg(v)
} 7→ {д
1
, ..., д
deg(u)
} such that L(f
i
) =
L(A(f
i
)).
4. Either ∀l
1
, l
2
∈ L(e)
2
, l
1
, l
2
: w
×
(l
1
, l
2
,v) ≤ w
×
(l
1
, l
2
, u), or ∀l
1
, l
2
∈ L(e)
2
, l
1
, l
2
:
w
×
(l
1
, l
2
, u) ≤ w
×
(l
1
, l
2
,v)
We call e the major leg, f
i
the left minor legs, and д
i
the right minor legs. Split u and v in
the same fashion as in Untangling Rule 2, but ensure that u
0
and v
0
, as well as u
00
and v
00
are adjacent to matching minor legs.
See Figure 4.31, right for an example.
We again have to store an ordering of e
0
and e
00
to later be able to reconstruct a line or-
dering solution for the original input graph. Again, these orderings directly correspond
to the partial ordering of L(e) induced by either the minor legs at u, or the minor legs at
v. But these partial orderings may not be compatible. If that was the case, the splitting
of e hides unavoidable crossings that will be later re-introduced during the reconstruc-
tion of the line ordering for the original input graph. To ensure that these crossings will
occur at the optimal position, we base the ordering of e
0
and e
00
on the partial ordering
induced by the minor legs at v if ∀l
1
, l
2
∈ L(e)
2
, l
1
, l
2
: w
×
(l
1
, l
2
,v) ≤ w
×
(l
1
, l
2
, u), or on
the partial ordering induced by the minor legs at v if otherwise.
Lemma 4.24. Untangling Rule 4 does not aect the optimality of the line ordering.
Proof. W.l.o.g., we assume ∀l
1
, l
2
∈ L(e)
2
, l
1
, l
2
: w
×
(l
1
, l
2
, u) ≤ w
×
(l
1
, l
2
,v). Per
Lemma 4.15, the minor legs at u induce a partial ordering on the L(e) which does not
induce any crossings or separations at u. Similarly, the minor legs at v induce a partial
142 Chapter 4. Line Ordering Optimization
v
{A, B}
{C}
{A, B, C}
д
f
e
{A, B}
д
0
{C}
u
v
{A, B, D}
{C, D, E}
h
i
{A, B}
{C}
e
0
e
00
f
0
v
0
v
00
u
{A, B, D}
{C, D, E}
h
i
j
{E}
j
{E}
Figure 4.33: Illustration of Untangling Rule 5.
ordering on the L(e) which does not induce any crossings or separations at v. There are
now two cases: (1) The ordering induced at u is exactly inverse to the ordering induced
at v (see Figure 4.32, left for an example). (2) The ordering induced at u is not inverse to
the ordering induced at v (see Figure 4.31, left for an example). In case (1), the two order-
ings match, and any optimal line ordering solution will have no crossings or separations
between two lines l from dierent minor legs, and no optimal crossing or separation is
prohibited by Untangling Rule 4. In case (2), the induced partial ordering will also not
add any separations at u or v between two lines from dierent minor legs, as no two
such lines continue together from e over u or v. The inter-partion crossings induced
in case (2) are unavoidable and must occur either at u, or at v. Implicitly moving them
to v by an appropriate ordering of the split edges e
0
and e
00
is optimal, as we assumed
∀l
1
, l
2
∈ L(e)
2
, l
1
, l
2
: w
×
(l
1
, l
2
, u) ≤ w
×
(l
1
, l
2
,v).
Figure 4.33, left shows the Double-Y equivalent of a partial Y structure, a partial
Double-Y structure, which may be untangled in a similar way as partial Y structures:
Untangling Rule 5 (Partial Double Y). Given an edge e = {u,v} with deg(u) ≥ deg(v) ≥
3 and the following additional properties:
1. Each l ∈ L(e) uniquely extends over v into one of n < deg(v) edges f
1
, ..., f
n
, f
i
, e,
and all l ∈ L(f
i
)∩L(e) uniquely extend over u into e (that is, L(f
1
)∪... ∪L(f
n
) = L (e)
and L(f
1
) ∩ ... ∩L (f
n
) ∩ L(e) = ∅).
2. Each l ∈ L (e) uniquely extends over u into one of deg(u)−1 edges д
1
, ..., д
deg(v)
, д
i
, e,
and all l ∈ L(д
i
) uniquely extend over v into e (that is, L(д
1
)∪... ∪L(д
deg(v)−1
) = L (e)
and L(д
1
) ∩ ... ∩L (д
deg(v)−1
) = ∅).
3. There is an injective mapping A : {д
1
, ..., д
deg(u)
} 7→ {f
1
, ..., f
deg(v)
} such that L(д
i
) ⊆
L(A(д
i
)).
4. ∀l
1
, l
2
∈ L(e)
2
, l
1
, l
2
: w
×
(l
1
, l
2
,v) ≤ w
×
(l
1
, l
2
, u).
Split v in the same fashion as in Untangling Rule 4, but leave u as is.
Note that the only dierence to Untangling Rule 5 is that we allow additional edges
at v, as well as lines extending over v from minor legs.
4.9 Line Graph Simplication 143
v
{C, B}
{A, B, C, D, E}
f
e
u
{A, B, F }
{C, F }
h
i
д
{D, E}
j
{A}
v
{C, B}
{A, B, C}
f
e
u
{A, B, F }
{C, F }
h
i
д
{D, E}
j
{A}
u
0
v
0
{D, E}
e
0
Figure 4.34: Illustration of Untangling Rule 6.
Lemma 4.25. Untangling Rule 5 does not aect the optimality of the line ordering.
Proof. The correctness proof is analogous to that of Untangling Rule 5.
Stump Structures
We now consider situations like depicted in Figure 4.34. It is easy to see that if all lines
L(i) on a minor leg i uniquely extend over u into a main leg e = {u, v}, if i and e are next
to each other in the circular edge ordering at u, and if all L(i) have a terminus at v, then
we may always place all L(i) on the “outside” of e without any crossings or separations
between a line l
1
∈ L(i), and a line l
2
< L(i). We say i forms an outer stump in e.
The following untangling rule captures this intuition:
Untangling Rule 6 (Outer Stump). Given an edge e = {u, v} with deg(u) > 2 and
deg(v) ≥ 2 and an edge i = {w, u}. Additionally, the following properties hold:
1. Each l ∈ L (i) uniquely extends over u into e.
2. Each l ∈ L (i) terminates at v.
3. L(i) , L(e)
4. Some line m ∈ L(e) \ L(i) extends over v.
5. Some line m ∈ L(e) \ L(i) extends over u.
6. Either π
v
e
(i) = 1 or π
v
i
(e) = 1.
Detach the stump by splitting u and v like depicted in Figure 4.34.
Lemma 4.26. Untangling Rule 6 does not aect the optimality of the line ordering.
Proof. As all L(i) have a terminus at v, we may place them either at the left or right side
of the ordering without introducing any crossings or separations in v. If π
v
e
(f ) = 1,
placing them on the right side w.r.t. u will induce no crossings or separations between
a line l
1
∈ L(i) and a line l
2
< L(i) at u, and if π
v
f
(e) = 1, placing them on the left side
w.r.t. u will induce no such crossings or separations at u. The partial ordering induced
by Untangling Rule 6 will thus always be optimal.
144 Chapter 4. Line Ordering Optimization
v
{A, B}
{A, B, C, D}
f
e
u
{A, B}
{C}
h
i
v
{D}
f
e
u
{A, B}
h
{A, B, C, D}
д
{D}
j
{D}
{C}
i
д
{D}
j
{A, B}
{C}
k
w
Figure 4.35: Illustration of Untangling Rule 7.
A dierent kind of stump which we call inner stump is depicted in Figure 4.35 and
may be transformed into a situation eligible for Untangling Rule 4 or 5 as follows:
Untangling Rule 7 (Inner Stump). Given an edge e = {u, v} which would fulll every
requirement for a full or partial double Y, but there are minor left legs S
l
whose lines end at
v (but not at u), and/or minor right legs S
r
whose lines end at u (but not at v), and which
are also not eligible for Untangling Rule 6 (they are inner stumps). Let A : adj(u) \ {e} \
S
r
7→ adj(v) \ {e} \S
l
be a bijective mapping which assigns each left minor leg f a right
minor leg with L(f ) = L(A(f )). If the circular edge orderings w.r.t e of the left non-stump
minor legs and their corresponding right non-stump minor legs are exactly inverse, add
imaginary extensions for u or v for each stump leg as depicted in Figure 4.35, and proceed
as in Untangling Rule 4.
Edge e is now eligible for Untangling Rule 4.
Lemma 4.27. Untangling Rule 7 does not aect the optimality of the line ordering.
Proof. As the circular edge orderings w.r.t. e of the non-stump minor legs are exactly
inverse, they induce a matching partial ordering without any crossings or separations
between two lines l
1
and l
2
from dierent minor non-stump legs. If we extend the left
and right minor legs as described, the induced partial orderings still match on both sides.
The partial ordering induced by the subsequent application Untangling Rule 4 is thus
optimal.
A very simple case not yet covered by any of the previous is the following:
Untangling Rule 8 (Double Stump). Given an edge e = {u, v} which is not eligible for
Cutting Rule 2 (Terminus Detachment)) and on which one or more lines l ∈ L(e) have a
terminus at u and v. Delete l from L(e) and settle its nal position to be at the outside of e.
Lemma 4.28. Untangling Rule 7 does not aect the optimality of the line ordering.
Proof. This follows from the fact that l cannot induce any crossing at u or v, and if placed
at the outside of e, can also not induce any separation.
4.9 Line Graph Simplication 145
1
10
5
5
4
6
1
1
{𝐴, 𝐵, 𝐶 }
𝑣
1
𝑣
2
𝑣
3
𝑣
4
1
10
5
5
1
1
𝑣
1
𝑣
2
𝑣
4
{𝐴, 𝐵, 𝐶 }
{𝐴, 𝐵, 𝐶 }
{𝐴, 𝐵, 𝐶 }
{𝐴, 𝐵, 𝐶 }
Figure 4.36: A single round of Pruning Rule 1 (Node Contraction) is applied on a line
graph G with nodes v
1
, . . . .v
4
in the order given by the node indices (from left to right).
The crossing and separation penalties between lines A and B in each node are depicted
inside the node. During the rst round, only v
3
is contracted, but v
2
is now eligible for
contraction.
4.9.5 Full Simplication
Naturally, our interest is to combine the pruning, cutting, and untangling rules described
so far and apply them as often as possible. This gives rise to several questions, in partic-
ular: (1) How often can a simplication rule be applied? (2) Which simplication rules
may create new situations eligible for other simplication rules? (3) Is there an adverse
order of applying the rules?
This section aims to answer these questions. In particular, we determine upper
bounds on the number of times a simplication rule may be applied, and proof several
lemmas on their application order. The insight gained is then used to give an algorithm
which performs a full simplication (that is, after the algorithm is applied, none of the
simplication rules can be applied anymore) with a bounded number of iterations.
We call the standalone application of a simplication rule on all original nodes (Prun-
ing Rule 1, Cutting Rule 2), all original lines (Pruning Rule 2), or all original edges (Prun-
ing Rule 3, Cutting Rule 1, and Untangling Rules 1- 8) a round of that rule.
In the general MLNCM-WS case, a single round of Pruning Rule 1 (Node Contrac-
tion), applied by simply iterating over each node v ∈ |V |, might result in some remaining
nodes eligible for contraction. This happens for simple paths on which all edges have
the same lines, and on which all inner nodes have degree 2, as depicted in Figure 4.36,
and depends on the ordering in which the nodes are contracted. Observe, however, that
such a situation might only occur locally between two nodes that were separated by a
previously contracted degree-2 node. For Pruning Rule 1 (Node Contraction), we there-
fore apply a full round like this: apply Pruning Rule 1 (Node Contraction) to each node
|V |. If a node v is contracted and was previously adjacent to a node u with deg(u) = 2,
immediately apply Pruning Rule 1 (Node Contraction) to u (there are at most 2 such
nodes, as deg(v) = 2). We then still process only O(|V |) nodes, but now no opportunity
for contraction remains after a single round of Pruning Rule 1 (Node Contraction).
Lemma 4.29. The order in which we process the lines for Pruning Rule 2 (Line Partner
Collapse), the edges for Pruning Rule 3 (Single Edge Prune), or the nodes for Cutting Rules 1
(Single Line Cut) and 2 (Terminus Detachment) in a single standalone round of the respective
rule is irrelevant.
146 Chapter 4. Line Ordering Optimization
Proof. (1) If Pruning Rule 2 (Line Partner Collapse) would collapse lines B into a single
new line k, then any line set C that would have been eligible for a collapse with any
subset of B is also eligible for a collapse with {k}. (2) Each edge e eligible for Pruning
Rule 3 (Single Edge Prune) is its own graph component, its deletion does not aect other
edges. (3) Cutting Rule 1 (Single Line Cut) may neither destroy a new situation eligible
for Cutting Rule 1 (Single Line Cut), nor create one (the cut edges had at least one adja-
cent node with degree 1). (4) Cutting Rule 2 (Terminus Detachment) applied on a node v
may neither add, nor remove an opportunity for Cutting Rule 2 (Terminus Detachment)
at a node w , v.
The following lemma proves that we might also never “destroy” an opportunity for
pruning and cutting by an adverse order of application of dierent types of cutting or
pruning rules.
Lemma 4.30. Given a line graph G and either some node v eligible for Cutting Rule 2
(Terminus Detachment) or Pruning Rule 1 (Node Contraction), or some edge e eligible for
Pruning Rule 3 (Single Edge Prune) or Cutting Rule 1 (Single Line Cut). It holds that no
application of another pruning or cutting rule might result in a line graph G
0
in which v
(or e) are not eligible anymore for the respective rule.
Proof. None of the cutting rules requires a minimum number of lines on an edge, so
Pruning Rule 2 (Line Partner Collapse) will not destroy any cutting opportunity. Cutting
Rule 1 (Single Line Cut) never removes any lines, never adds any lines, never detaches
a line from some node v, and never attaches an edge to a node with degree 2, so it will
never remove an opportunity for Pruning Rule 3 (Single Edge Prune) or Cutting Rule 2
(Terminus Detachment). Cutting Rule 2 only detaches an edge e not eligible for Cutting
Rule 1 (Single Line Cut) from a node v not eligible for Cutting Rule 1 (Single Line Cut),
both by denition, and does not aect any other node or edge. Pruning Rule 2 (Line
Partner Collapse) only removes nodes not eligible for any other pruning or cutting rule
by denition, and leaves line adjacencies at other nodes intact. Pruning Rule 3 (Single
Edge Prune) only operates on graph components not eligible for any other pruning or
cutting rule per denition.
Lemma 4.31. When applied in rounds, Pruning Rule 2 (Line Partner Collapse), Pruning
Rule 3 (Single Edge Prune) and Cutting Rules 1-2 are idempotent.
Proof. (1) Consider a line graph G after a round of Pruning Rule 2 (Line Partner Col-
lapse). If there is a line k which is the result of a line partner collapse and which is now
eligible for Pruning Rule 2 (Line Partner Collapse) with a line l, then l would’ve already
been eligible for Pruning Rule 2 (Line Partner Collapse) with any line combined in k, or
even with k itself. (2) Pruning Rule 3 (Single Edge Prune) does not create new terminus
4.9 Line Graph Simplication 147
nodes. (3) An edge resulting from a cut by Cutting Rules 1 (Single Line Cut) or 2 (Termi-
nus Detachment) is never eligible for Cutting Rules 1 (Single Line Cut) or 2 (Terminus
Detachment), as one of its adjacent nodes has now degree 1.
Lemma 4.32. Given a line graph G with no situation where Pruning Rule 2 (Line Partner
Collapse) may be applied. No cutting, pruning, or untangling rule may result in a line graph
G
0
containing a situation where Pruning Rule 2 (Line Partner Collapse) may be applied.
Proof. There are two cases to consider. (1) There are no two lines l
1
and l
2
which follow
the same path through G, and a cutting, pruning, or untangling rule would create such
two lines. (2) Two lines l
1
and l
2
follow the same path p, but p is not simple, and a cutting,
pruning, or untangling rule would make p simple. Regarding (1), no cutting, pruning,
or untangling rule adds a line to an edge or collapses two edges into one. Additionally,
no cutting, pruning, or untangling rule detaches an edge e from a node v over which
a line l ∈ L (e) extends, so no such rule could detach an edge in such a way that a line
l
1
would now follow the same path as a line l
2
(Cutting Rule 2 (Terminus Detachment)
does only cut an edge, not detach). For the same reason, no such rule could transform a
non-simple path p of two lines l
1
and l
2
into a simple path.
Lemma 4.33. Given a line graph G with no situation where Cutting Rule 1 (Single Line
Cut) may be applied. A single round of Cutting Rule 2 (Terminus Detachment) may never
result in a line graph G
0
containing a situation where Cutting Rule 1 (Single Line Cut) may
be applied.
Proof. Cutting Rule 2 (Terminus Detachment) never decreases the number of lines.
The same is true in the other direction:
Lemma 4.34. Given a line graph G with no situation where Cutting Rule 2 (Terminus
Detachment) may be applied. A single round of Cutting Rule 1 (Single Line Cut) may never
result in a line graph G
0
containing a situation where Cutting Rule 2 (Terminus Detachment)
may be applied.
Proof. Cutting Rule 1 (Single Line Cut) only introduces terminus nodes at the newly
inserted cut nodes, but these have degree 1.
Lemma 4.35. Given a line graph G with no situation where a cutting rule may be applie d.
A single round of Pruning Rule 1 (Node Contraction) may never result in a line graph G
0
containing a situation where a cutting rule may be applied.
Proof. Pruning Rule 1 (Node Contraction) never decreases the number of lines per edge,
so Cutting Rule 1 (Single Line Cut) may not suddenly be applicable. It also does not intro-
duce terminus nodes, so Cutting Rule 2 (Terminus Detachment) may also not suddenly
be applicable.
148 Chapter 4. Line Ordering Optimization
v
{A, B, D}
{C}
{A, B, C}
f
д
e
e
0
e
00
u
u
0
u
00
v
f
д
h
{D}
h
{A, B, D}
{C}
{A, B}
{D}
{C}
{A, B}
{C}
{A, B, C}
f
00
д
e
u
h
{D}
w
w
w
0
f
0
v
w
00
{D}
{A, B}
{C}
f
00
д
h
{D}
w
0
f
0
w
00
{D}
e
0
e
00
u
0
u
00
{A, B}
{C}
v
Figure 4.37: Illustration of the proof for Lemma 4.38.
Lemma 4.36. Given a line graph G with no situation where any cutting or pruning rule,
except Pruning Rule 3 (Single Edge Prune), may be applied. A single round of Pruning Rule 3
(Single Edge Prune) may never result in a line graph G
0
where any pruning or cutting rule
may be applied.
Proof. This follows from the fact that each pruned edge is in its own component.
Corollary 4.37. Algorithm 4.1 exhaustively prunes and cuts a line graph G.
Next, we investigate whether there is an adverse application order of the untangling
rules, or whether the untangling rules might destroy opportunities for cutting or pruning
(or vice versa).
Lemma 4.38. The order in which a single standalone round of any Untangling Rule is
performed (that is, the ordering of the edges or nodes we perform each rule on) is irrelevant.
Proof. (1) An application of Untangling Rule 1 (Full X) on some node v does not aect any
other node, so we only have to consider v itself. Assume we detached an edge pair e, f ,
and there is an edge д who would’ve also been eligible for Untangling Rule 1 together
with e (or f ). But then the lines on L(e) branched at v into f and д, and e and f would
not have been eligible for Untangling Rule 1. (2) All other untangling rules always split
up a major leg edge e, but leave the minor legs intact. For a full Y, a full double Y, a
minor leg edge f could not have been a major leg edge with e as a minor leg edge by
denition. (If there was another edge д which, as a major leg, would’ve been eligible for
some untangling rule with f as a minor leg, then f , or an edge f
0
adjacent to д and with
4.9 Line Graph Simplication 149
G ←− Pru ninдRule[2](G);
G ←−CuttinдRule[1](G);
G ←−CuttinдRule[2](G);
G ←− Pru ninдRule[1](G);
G ←− Pru ninдRule[3](G);
Algorithm 4.1: Full pruning and cutting of a line graph G.
the same lines, are still present after applying the rule on e). For a partial Y, a partial
double Y, or stump structures, if two adjacent edges e and f with L(e) , L(f ) share a set
of lines A = L(e)∩L(f ), e might be the minor leg for f , and f might be a minor leg for
e (see Figure 4.37). But in such a case, the untangling opportunity on e is still present
after f is untangled, and the application order does therefore not matter.
Similar as for the pruning and cutting rules, the following lemma shows that we
might never destroy an untangling opportunity by an adverse order of application of
dierent types of untangling rules.
Lemma 4.39. Given a line graph G and either some node v eligible for Untangling Rule 1
(Full X), or some edge e eligible for Untangling Rules 2 - 7. It holds that no application of
another untangling rule might result in a line graph G
0
in which v (or e) are not eligible
anymore for the respective rule.
Proof. For Untangling Rules 2 - 7, the argument is the same as (2) in the proof for
Lemma 4.38. Untangling Rule 1, may also never destroy an opportunity for Untangling
Rules 2 - 7, as two edges e and f eligible for Untangling Rules 1 can never be a minor
or major leg for Untangling Rules 2 - 7. The same argument holds for the other direc-
tion, and Untangling Rules 2 - 7 may also never destroy an opportunity for Untangling
Rule 1.
Most importantly, we might also never destroy an untangling opportunity by a prun-
ing or cutting rule:
Lemma 4.40. Given a line graph G and either some node v eligible for Untangling Rule 1
(Full X), or some edge e eligible for Untangling Rules 2 - 7. It holds that no application of
any pruning or cutting rule might result in a line graph G
0
in which v (or e) are not eligible
anymore for the respective rule.
Proof. Let two edges e and f adjacent at v eligible for Untangling Rule 1. By denition,
v cannot be eligible for Pruning Rule 1 (Node Contraction), e and f will still be eligible
after Pruning Rule 2 (Line Partner Collapse) has collapsed lines on them, Pruning Rule 3
(Single Edge Prune) cannot be applied, Cutting Rule 1 (Single Line Cut) may create novel
150 Chapter 4. Line Ordering Optimization
edges e
0
and f
0
still eligible for Untangling Rule 1 at v, and Cutting Rule 2 by denition
cannot destroy the eligibility for Untangling Rule 1. Each remaining untangling rule
requires a major leg e with |L(e)| > 1, whose lines branch at one (or two) adjacent nodes
v (or v,u) into minor legs, rendering the major leg e ineligible for any cutting rule, and
v, u ineligible for any pruning rule involving major and/or minor legs.
Last, we might also never destroy an opportunity for pruning or cutting by an un-
tangling rule:
Lemma 4.41. Given a line graph G and either some node v eligible for Cutting Rule 2 (Ter-
minus Detachment) or Pruning Rule 1 (Node Contraction), some edge e eligible for Pruning
Rule 3 (Single Edge Prune) or Cutting Rule 1 (Single Line Cut), or some set of lines B eligible
for Pruning Rule 2 (Line Partner Collapse). It holds that no application of an untangling
rule might result in a line graph G
0
in which v, e or B are not eligible anymore for the
respective rule.
Proof. Regarding Cutting Rule 1 (Single Line Cut), no untangling rule adds lines or in-
troduces terminus nodes. Regarding Cutting Rule 2 (Terminus Detachment), no untan-
gling rule adds lines or detaches a terminus edge. Regarding Pruning Rule 1 (Node Con-
traction), untangling rules also never increase the degree of degree-2 nodes. Regarding
Pruning Rule 3 (Single Edge Prune), no untangling rule is eligible for such a single edge
structure. Regarding Cutting Rule 2 (Line Partner Collapse), two line sets A and B eli-
gible for a line partner collapse can never be split by an untangling rule, as they never
branch (by denition).
For the untangling rules, we also have to consider the fact that the application of a
single round of any untangling rule, and the subsequent application of Algorithm 4.1,
might create a new situation again eligible for an untangling rule. An example of such a
situation is given in Figure 4.38. It might also create new situations eligible for cutting
and pruning rules. We therefore have to keep applying all simplication rules in the
order given in Algorithm 4.2 until the graph does indeed not change anymore. By the
lemmas proved above, we can then be sure that all simplication opportunities have been
used, and that we couldn’t have arrived at a “simpler” graph with another application
order. The following corollary captures this:
Corollary 4.42. Algorithm 4.2 exhaustively prunes, cuts, and untangles a line graph G.
Next, we determine an upper bound of the number of outer loop iterations of Algo-
rithm 4.2.
Lemma 4.43. Given an initially fully pruned and cut line graph G to which a single outer
loop iteration of Algorithm 4.2 has been applied. If in the next iteration Untangling Rule 1
(Full X) is applicable, then one (or both) minor leg edges resulted from a major leg split by
one of Untangling Rules 2-7 in the previous iteration.
4.9 Line Graph Simplication 151
𝑣
{𝐴, 𝐵}
{𝐴, 𝐵, 𝐶, 𝐷}
𝑢
{𝐴, 𝐵}
{𝐶, 𝐷}
{𝐵}
{𝐴}
{𝐵}
{𝐴}
𝑣
0
{𝐴, 𝐵}
{𝐴, 𝐵}
𝑢
0
{𝐴, 𝐵}
{𝐵}
{𝐴}
{𝐵}
{𝐴}
𝑢
00
𝑣
00
{𝐶, 𝐷}
{𝐶, 𝐷}
{𝐶, 𝐷}
{𝐶, 𝐷}
{𝐴, 𝐵}
{𝐵}
{𝐴}
{𝐵}
{𝐴}
{𝐶, 𝐷}
{𝐵}
{𝐴}
{𝐵}
{𝐴}
{𝐶, 𝐷}
{𝐵}
{𝐴}
𝑤
𝑤
𝑤
𝑡
𝑡
𝑡
𝑡
0
𝑡
00
𝑤
𝑤
0
𝐺
𝐺
0
𝐺
00
𝐺
000
Figure 4.38: Untangling opportunities created by an untangling rule.
Proof. All initial opportunities for Untangling Rule 1 have been resolved in the rst iter-
ation of Algorithm 4.2, and no application of Untangling Rule 1 or any pruning or cutting
rule may create a novel opportunity for Untangling Rule 1. The novel opportunity was
therefore triggered by one of untangling rules 2 - 7. Assume that a minor leg edge of
Untangling Rules 2-7 triggered the opportunity. But Untangling Rules 2-7 never change
any minor leg edges, and therefore the opportunity must have been created by a split of
a major leg edge.
Lemma 4.44. Given an initially fully pruned and cut line graph G to which a single outer
loop iteration of Algorithm 4.2 has been applied. If in the next iteration there is a new
situation eligible for untangling rules 2 - 7, then the line bundle of the major leg of that rule
was part of a major leg in the previous iteration.
Proof. We prove this by contradiction and assume it is not case. Then there are three
possibilities: (1) the novel opportunity was created by a combination of pruning and
cutting rules, without use of any untangling rule. (2) the major leg resulted from two
minor legs of Untangling Rule 1, without use of any other untangling rule. (3) the major
leg was already present, and a new minor leg enabling the application of the untangling
rule was created by untangling rules 2 - 7 in the last iteration.
Regarding (1), we assumed the graph to be initially fully pruned and cut. A stan-
dalone application of the pruning and cutting rules would thus not have altered the
152 Chapter 4. Line Ordering Optimization
G ←− Pru ninдRule[2](G);
do
G
∗
←− G;
G ←−U ntanдlinдRule[2 −6](G);
G ←−U ntanдlinдRule[1](G);
G ←−CuttinдRule[1](G);
G ←−CuttinдRule[2](G);
G ←− Pru ninдRule[1](G);
G ←− Pru ninдRule[3](G);
while G
∗
, G;
Algorithm 4.2: Full pruning, cutting and untangling of a line graph G, with explicit
check if the graph is fully simplied.
input graph. Regarding (2), Lemma 4.43 states that the minor leg then resulted from a
major leg of untangling rules 2 - 7. Regarding (3), if a minor leg edge f for untangling
rules 2 - 7 resulted from the split of a major leg edge in the previous iteration, this would
either mean that a minor leg edge with the same lines (and therefore the untangling op-
portunity) was already present in the previous iteration, or that f was already eligible
for untangling in the previous iteration (see the example in Figure 4.37).
Lemma 4.45. Given an initially fully pruned and cut line graph G. Then the number of
iterations of Algorithm 4.2 is bounde d by M −1, where M is the maximum number of lines
per edge in G.
Proof. This directly follows from Lemmas 4.43 and 4.44. At each iteration, the minor
leg(s) of Untangling Rule 1 or the major legs of Untangling Rule 2-7 resulted from a
split of a major leg edge in the previous iteration. For each application of an untangling
rule, there is thus a path of previous major leg splits back to the initial input state of
Algorithm 4.2, and this path has a maximum length of M − 1 (a set of M lines can only
be split M − 1 times).
This is reected in Algorithm 4.3 (because we assumed the graph to be initially
pruned and cut, we use M iterations).
Corollary 4.46. Algorithm 4.3 exhaustively simplies a line graph G.
It remains to show that a full simplication may be performed in reasonable time.
The next section will show that it is possible to fully simplify any given input line graph G
in time polynomial only in the maximum number M of lines per edge, and the maximum
node degree D.
4.9 Line Graph Simplication 153
i ←− M;
G ←− Pru ninдRule[2](G);
while i > 0 do
G ←−U ntanдlinдRule[2 −6](G);
G ←−U ntanдlinдRule[1](G);
G ←−CuttinдRule[1](G);
G ←−CuttinдRule[2](G);
G ←− Pru ninдRule[1](G);
G ←− Pru ninдRule[3](G);
i ←− i − 1;
end
Algorithm 4.3: Full pruning, cutting and untangling of a line graph G, with a guaran-
teed number of outer loop iterations.
4.9.6 Complexity
We rst examine the complexity of applying a single round of each pruning, cutting or
untangling rule on some graph G. Afterwards, we analyze the complexity of Algorithm
4.3.
By D = max
v∈V
deg(v) we again denote the maximum node degree of our input line
graph G. By M = max
e∈E
|L(e)| we again denote the maximum number of lines per
segment. We may remove nodes with degree 0 from the line graph prior to optimization
and thus assume that 2|E| is an upper bound for |V |.
For our analysis, we additionally make the following assumptions:
1. Each line is identied by a numerical ID.
2. Each edge is identied by a numerical ID.
3. Each node is identied by a numerical ID.
4. V is ordered by the node IDs (as is typically the case if nodes are stored in a con-
tinuous array).
5. For each e, L(e) is ordered by the line IDs
6. For each v, adj(v) is ordered by the outgoing angle of the adjacent edges.
7. For a line l ∈ L and an edge e, we can check whether l ∈ L(e) in constant time (for
example, by building a lookup array).
8. For a line l ∈ L and a node v, we can check whether v is a terminus for l in constant
time (for example, by storing this information per node).
9. We can retrieve the lines T
v
that terminate at a node v in constant time (for example,
by storing this information beforehand).
10. For an edge e = {u, v}, we can check in constant time whether for all l
1
, l
2
∈
L(e)
2
, l
1
, l
2
, w
×
(l
1
, l
2
, u) ≤ w
×
(l
1
, l
2
,v) or vice versa.
154 Chapter 4. Line Ordering Optimization
11. For an edge e = {u, v}, we can check in constant time whether for all l
1
, l
2
∈
L(e)
2
, l
1
, l
2
, w
||
(l
1
, l
2
, u) ≤ w
||
(l
1
, l
2
,v).
The last two points need some additional explanation. We can allow constant time
lookups of these properties if we initially determine and store tuples (w
min
×u
, w
max
×u
) and
(w
min
||u
, w
max
||u
) for both sides of all edges e = {u, v} ∈ E, where w
min
×u
is the minimum
cost w
×
(l
1
, l
2
, u) over all l
1
, l
2
∈ L(e)
2
, l
1
, l
2
, w
max
×u
is the maximum such cost, and so
on. These values can be determined in O(|E|M
2
) time. Then to determine if for exam-
ple w
×
(l
1
, l
2
, u) ≤ w
×
(l
1
, l
2
,v) for all l
1
, l
2
∈ L(e)
2
, l
1
, l
2
, it suces to check whether
w
max
×u
≤ w
min
×v
. When two edges are merged during the application of Pruning Rule 1
(Node Contraction), we update the tuples accordingly, which can be done in constant
time. Note that when an edge and an adjacent node v are split in one of the Untan-
gling Rules, an update is not required, as we must always consider the original input
node v
∗
for the crossing and separation weights to ensure optimality, as explained in
Section 4.9.2.
Complexity of Pruning Rules
For a single application of Pruning Rule 1 (Node Contraction) to some node v with
deg(v) = 2, we have to check whether adjacent edges e and f for a node v with deg(v) =
2 have L(e) = L(f ), and whether the crossing and separation weights at the remaining ad-
jacent nodes are equal or smaller. This can be done in O(M), as we assumed that (1) both
L(e) and L(f ) are sorted and (2) the check whether the crossing and separation weights
are smaller or equal can be done in constant time. In a single round of Pruning Rule 1
(Node Contraction) we have to contract O(|V |) nodes, as described in Section 4.9.5. A
single round of Pruning Rule 1 (Node Contraction) can therefore be applied in O(|E|M).
For Pruning Rule 2 (Line Partner Collapse), we use the following algorithm:
(1) Iterate over each v ∈ V (recall that V is ordered by the node IDs) (2) For each
l ∈ T
v
, mark l as processed and follow its path through G until the path either terminates,
diverges, or arrives at v again. Remember the visited node IDs as well as their crossing
and separation weights. If the path terminates, append the list of visited nodes (with
their crossing and separation weights) to a list A and continue. If the path was not
simple, drop it, and continue. (3) Sort A and extract the line clusters with equal path and
weights. (4) For each such line cluster B with |B| > 1, choose any l ∈ B as a reference
line, update the crossing weights for l and delete all B \ {l} from the line graph.
For steps (1) and (2) we have to follow at most |E| edges at most M times, so these
can be done in O(|E|M). A can be sorted in O(|E|M log M), and matching line clusters
can then be retrieved in O(|E|M). Finally, the line collapsing in step (4) can be done in
O(|E|M), so we need time O(|E|M log M) overall.
For Pruning Rule 3 (Single Edge Prune), we iterate over all edges, deleting edges
where both adjacent nodes have a degree of 2. This can be done in O(|E|).
4.9 Line Graph Simplication 155
Complexity of Cutting Rules
Cutting Rule 1 (Single Line Cut) has to check for each edge e whether |L(e)| = 1 and
then cut this edge, adding two new nodes and edges. This can be done in O(|E|).
As we assumed that we can check whether a line ends at a node in constant time,
we can retrieve all candidates for Cutting Rule 2 (Terminus Detachment) in O(|E|). The
edge detachment can then be done in constant time.
Complexity of Untangling Rules
To apply Untangling Rule 1 (Full X) to a single node, a naive way would be to test for
all pairs {e, f } of adjacent edges whether L(e) = L(f ). As there are at most
D
2
< D
2
such pairs, this can be done in O(MD
2
), resulting in a time complexity of O(|V |MD
2
) =
O(|E|MD
2
) for a full round of Untangling Rule 1 (Full X).
However, as we assumed L(e)to be sorted for each e, we may sort L(e
1
), . . . , L(e
deg(v)
)
for adjacent edges e
1
, . . . , e
deg(v)
in O(MD log D) and extract matching clusters of size 2
in O(MD), improving the complexity for a full round of Untangling Rule 1 (Full X) to
O(|V |MD log D) = O(|E|MD log D).
For applying Untangling Rule 2 (Full Y) to a single node, we have to identify a major
leg e (this has a time complexity of O(D)) and check whether the lines of the remaining
adjacent edges (the minor legs) are contained in L(e). This takes O(MD). To check
whether the minor legs f
1
, . . . , f
N
are pairwise disjoint, we rst do a k-way merge of the
L(f
i
) and then check whether |
L(f
i
)| =
|L(f
i
)|. This takes O(MD log D). Applying
Untangling Rule 2 (Full Y) to all v ∈ V thus takes O(|V |MD log D) = O(|E|MD log D).
Assume we apply Untangling Rule 3 (Partial Y) to a single node v, given a major leg
candidate e. We have to collect all adjacent edges f
i
, e which share a line with e, and
for each such f
i
check which lines are shared. This can again be done in O(MD) via a
simple list intersection. Checking whether these shared line sets are pairwise disjoint
can then again be done in O(MD log D) like for Untangling Rule 2 (Full Y). Applying
Untangling Rule 3 (Partial Y) to all v ∈ V therefore also takes O(|E|MD log D).
Double-Y structures may be understood as two compatible Single-Y structures, con-
nected by the bijective mapping m described in Untangling Rule 4 (Full Double Y). Ap-
plying Untangling Rule 4 (Full Double Y) to an edge e = {u, v} therefore amounts to
applying the check for Untangling Rule 2 (Full Y) for u and v (assuming that e as been
“cut”) and nding the bijective mapping m. The former can be done in O(MD log D) for
a single edge e. As we assumed both adj(u) and adj(v) to be sorted, the mapping m can
be found in O(MD) time and O(M) space by creating two arrays A
u
and A
v
of size M.
For each minor leg edge f
i
adjacent to u, we then take a random line l from L(f
i
) (as
the minor leg edges are pairwise disjoint, this is a unique identier), write the clockwise
position of f
i
to A
u
[l]. We do the same at v. Afterwards, we iterate over A
u
, check the
156 Chapter 4. Line Ordering Optimization
corresponding entry in A
v
for line equivalency and write the position pair to our map-
ping m. The complexity of applying Untangling Rule 4 (Full Double Y) to all v ∈ V is
therefore O(|E|MD log D).
For partial Double-Y structures, the argumentation is similar. W.l.o.g., applying Un-
tangling Rule 3 (Partial Y) to an edge e = {u, v} amounts to applying the check for
Untangling Rule 2 (Full Y) for u, applying the check for Untangling Rule 3 (Partial Y) to
v and nding the bijective mapping m. The complexity of applying Untangling Rule 3
(Partial Y) to all v ∈ V is therefore also O(|E|MD log D).
To collect all outer stumps for Untangling Rule 6 (Outer Stump) on a single edge e,
we have to check for four adjacent edges f
1
, f
2
, f
3
, f
4
whether they are minor legs for an
outer stump in the worst case. This can be done in O(M). A single round of Untangling
Rule 6 (Outer Stump) therefore has a complexity of O(|E|M).
Checking whether Untangling Rule 7 (Inner Stump) is applicable to a major leg edge
e has the same worst-time complexity as checking whether e is applicable for Untangling
Rule 4 (Full Double Y) or Untangling Rule 3 (Partial Y). The (imaginary) edge extension
can be done in constant time. Untangling Rule 7 (Inner Stump) therefore has the same
time complexity as Untangling Rules 7 (Inner Stump) and 4 (Full Double Y), which is
O(|E|MD log D).
Untangling Rule 8 (Double Stump) can be applied to a single edge e in time O(M) as
we assumed that we can check in constant time whether a line terminates at a node v.
Untangling Rule 8 (Double Stump) therefore has a complexity of O(|E|M).
Table 4.2 gives an overview of the complexities of each simplication rule.
Upper Bounds on Intermediate Graph Sizes
Algorithm 4.3 iteratively applies all simplication rules but Pruning Rule 2 (Line Partner
Collapse) to an intermediate graph G
∗
. Let
ˆ
M be the maximum number of lines, and
ˆ
D be
the maximum node degree of these graphs (including the original graph G). Similarly,
let
ˆ
N be the maximum number of edges of any graph.
As we never merge edges with dierent lines,
ˆ
M = M. The untangling rules might
split an edge and increase both the number of edges and the degree of an adjacent node,
but this may only happen M times per edge. In the worst case, the cutting rules might
double the number of edges. We therefore have
ˆ
D ≤ DM and
ˆ
N ≤ |E|2M.
Complexity of Full Simplication
Under the initial assumptions made above, the loop of Algorithm 4.3 therefore has a
worst-case asymptotic complexity of O(M
ˆ
N
ˆ
M
ˆ
D log
ˆ
D) = O(|E|M
4
D log MD). As we
have O(|E|M log M) ⊂ O(|E|M
4
D log MD), the single application of Pruning Rule 2 (Line
4.10 Full Solve Through Simplication 157
Table 4.2: Asymptotic complexities of single rounds of pruning, cutting, and untangling
rules
Rule Label Complexity
Pruning Rule 1 Node Contraction O(|E|M)
Pruning Rule 2 Line Partner Collapse O(|E|M log M)
Pruning Rule 3 Single Edge Prune O(|E|)
Cutting Rule 1 Single Line Cut O(|E|)
Cutting Rule 2 Terminus Detachment O(|E|)
Untangling Rule 1 Full X O(|E|MD log D)
Untangling Rule 2 Full Y O(|E|MD log D)
Untangling Rule 3 Partial Y O(|E|MD log D)
Untangling Rule 4 Full Double Y O(|E|MD log D)
Untangling Rule 5 Partial Double Y O(|E|MD log D)
Untangling Rule 6 Outer Stump O(|E|M)
Untangling Rule 7 Inner Stump O(|E|MD log D)
Untangling Rule 8 Double Stump O(|E|M)
Partner Collapse) does not change the asymptotic complexity of Algorithm 4.3, which
therefore remains O(|E|M
4
D log MD).
For all real-world datasets, both M and D are usually very small (in our biggest
dataset, M was 9, and D was 4) and may be assumed constants. Under this additional
assumption, Algorithm 4.3 runs in linear time.
4.10 Full Solve Through Simplication
For some line graph instances, applying the full simplication described above already
yields an optimal line ordering. This section describes several (non-trivial) classes of line
graph instances for which this is the case.
4.10.1 Tree-Like Line Graphs
We rst dene the following type of line graph, for which Figure 4.39 gives an example.
Denition 4.7 (Tree-Like Line Graph). A line graph G = (V , E, L, L) is called tree-like
if the following holds: (1) G is a tree. (2) Each l ∈ L follows a simple path through G. (3)
Each l ∈ L terminates at the root node. (4) Each l ∈ L terminates at a leaf node.
158 Chapter 4. Line Ordering Optimization
{A, B}
{C}
{A, B, C}
{A}
{B}
a
b
d
c
e
f
{C}
{A}
{B}
a
00
a
a
0
d
e
f
{A, B}
{C}
{A}
{B}
a
d
c
e
f
a
0
Figure 4.39: A tree-like line graph G is simplied into a forest of single-edge trees by
Algorithm 4.3, which are pruned by Pruning Rule 3 (Single Edge Prune), but displayed
here with dashed lines for better understanding. Algorithm 4.3 therefore already nds
an optimal line ordering for such instances.
Lemma 4.47. Given a tree-like line graph G with uniform crossing and separation weights.
Algorithm 4.3 yields an empty line graph G
0
(and thus an optimal line ordering for G).
Proof. The initial pruning will remove any two lines l
1
, l
2
with the same path through
G, as well as any inner nodes of degree 2. We can thus ignore these cases and focus on
trees with inner nodes of degree greater than 2, with no two lines following the same
path. Then two cases remain: (1) The root node r has degree 1. (2) The root node r has
a degree greater than 1. In the second case, as all lines terminate at the root, Cutting
Rule 2 (Terminus Detachment) will detach all edges from r, leaving a forest of deg(r)
trees with roots of degree 1. We therefore only have to consider case (1). Given then
such a tree-like line graph G with root r and deg(r) = 1, Untangling Rule 2 (Full Y) may
immediately be applied to the only child node v of r, leaving a forest of deg(v) tree-like
line graphs. This process continues until the tree has been untangled into a forest of
single-edge graphs, pruned by Pruning Rule 3 (Single Edge Prune). Figure 4.39 gives an
example.
Corollary 4.48. Given a tree-like line graph G with uniform crossing and separation
weights. An optimal MLNCM-WS solution for G can be found in O(|E|M
4
D log MD).
Corollary 4.49. Given a tree-like line graph G. An optimal MLNCM-S solution for G can
be found in O(|E|M
4
D log MD).
Corollary 4.50. Given a tree-like line graph G uniform crossing weights. An optimal
MLNCM-W solution for G can be found in O(|E|M
4
D log MD).
Corollary 4.51. Given a tree-like line graph G. An optimal MLNCM solution for G can be
found in O(|E|M
4
D log MD).
4.10 Full Solve Through Simplication 159
{𝐴, 𝐵}
{𝐶}
{𝐴, 𝐵, 𝐶}
{𝐴}
{𝐵}
𝑏
𝑑
𝑐
𝑒
𝑓
𝑒
𝑑
𝑒
𝑓
𝑒
{𝐴, 𝐵}
{𝐶}
{𝐴}
{𝐵}
𝑎
𝑔
ℎ
𝑗
𝑖
𝑔
𝑗
𝑖
{𝐴}
{𝐵}
{𝐶}
𝑟
{𝐴, 𝐵, 𝐶}
{𝐴, 𝐵}
{𝐶}
{𝐴}
{𝐵}
𝑏
𝑑
𝑐
𝑒
𝑓
𝑒
𝑒
{𝐴, 𝐵}
{𝐶}
{𝐴}
{𝐵}
𝑎
𝑔
ℎ
𝑗
𝑖
{𝐴, 𝐵, 𝐶}
{𝐴, 𝐵}
{𝐶}
{𝐴}
{𝐵}
𝑏
0
𝑑
𝑐
𝑒
𝑓
𝑒
{𝐴, 𝐵}
{𝐶}
{𝐴}
{𝐵}
𝑎
0
𝑔
ℎ
𝑗
𝑖
𝑏
𝑎
{𝐴, 𝐵}
{𝐶}
Figure 4.40: Simplifying a two-sided tree-like line graph (with root r ) by rst applying
Pruning Rule 1 (Node Contraction), and then Untangling Rule 2 (Full Y). This results in
a forest of two-sided tree-like line graphs, which can be similarly simplied.
4.10.2 Two-Sided Tree-Like Line Graphs
Denition 4.8 (Two-Sided Tree-Like Line Graphs). A line graph G = (V , E, L, L) is
called two-sided tree-like (2-tree-like) if the following holds: (1) G is a tree. (2) The root
of G has degree 2, the tree under the left root child is the left subtree, the tree under the
right root child is the right subtree. (3) Each l ∈ L follows a simple path through G. (4)
Each l ∈ L continues through the root node. (5) Each l ∈ L has a terminus at a leaf
node of the left subtree, and a terminus at a leaf node of the right subtree. (6) The left
and right subtree are branch-symmetric: for each branch of two or more line bundles at
level n in the left subtree there is a corresponding node in the right subtree where the
same line bundles branch.
Lemma 4.52. Given a 2-tree-like line graph G with uniform crossing and separation weights.
Applying Algorithm 4.3 to G will yield an empty line graph G
0
(and thus an optimal line
ordering for G).
Proof. The root node r will be initially contracted by Pruning Rule 1 (Node Contraction),
leaving an edge e to which Untangling Rule 2 (Full Y) can be applied. This leaves a forest
of double-sided tree-like line graphs, which are either linear with every edge sharing
the same set of lines, or again a non-trivial instance of a two-sided tree-like line graph.
In the former case, the line graph will be pruned away, in the latter case, the process
recursively continues (Figure 4.40).
Corollary 4.53. Given a 2-tree-like line graph G with uniform crossing and separation
weights. An optimal MLNCM-WS solution for G can be found in O(|E|M
4
D log MD).
160 Chapter 4. Line Ordering Optimization
Corollary 4.54. Given a 2-tree-like line graph G. An optimal MLNCM-S solution for G
can be found in O(|E|M
4
D log MD).
Corollary 4.55. Given a 2-tree-like line graph G uniform crossing weights. An optimal
MLNCM-W solution for G can be found in O(|E|M
4
D log MD).
Corollary 4.56. Given a 2-tree-like line graph G. An optimal MLNCM solution for G can
be found in O(|E|M
4
D log MD).
4.10.3 Real-World Examples
Two-sided tree-like line graphs are not only of theoretical interest, but are for example
often found in S-Bahn systems. These systems typically feature a central main corridor
which bundles all lines, which then branch into the suburbs on both sides of the corridor.
Figure 4.41, shows the ocial 2021 network map of the S-Bahn system in Stuttgart,
for which Algorithm 4.3 nds an optimal line-ordering solution with 2 crossings and 0
separations if the separation and crossing weights are uniform.
4.11 Experimental Evaluation
We have implemented all the methods described above: the full exhaustive search (Exh),
the greedy approach described in Section 4.6.3 (Gr), the greedy approach with lookahead
described in Section 4.6.3 (GrLa), GrLa with subsequent hillclimbing (GrLa+Hill), GrLa
with subsequent simulated annealing (GrLa+Ann), both simulated annealing and hill
climbing on a random initial line ordering (Hill, Ann), and the baseline ILP formulation
(b-ILP), and the improved ILP formulation (i-ILP). We tested them on 7 network datasets
of increasing complexity (see Table 4.3 for details): the 2016 tram network of Freiburg,
the 2016 light-rail network of Dallas, the 2016 light-rail network of Chicago, the 2015
(before heavy construction started) light-rail network of Stuttgart, the 2020 light-rail
network of Sydney, the 2016 tram network of Turin, and the 2016 subway network of
New York. Note that the Stuttgart network evaluated in this section is the light rail
network, not the (much simpler) S-Bahn network given in Figure 4.41. The datasets
were generated from raw GTFS data by our approaches from Chapters 2 and 3. Artifacts
left during this process were manually xed to ensure a fair comparison. For the New
York subway network, we increased the network complexity by explicitly considering
express lines (not done in the ocial network map). Evaluations were run on an Intel
Xeon X5560 machine with 8 cores (each with 2.8 GHz) and 36 GB of RAM.
For each dataset and method, we evaluated the weighted setting without punishing
line separations (MLNCM-W) and the weighted setting with punishing line separations
(MLNCM-WS). If line separations were considered, the methods are given in a starred
4.11 Experimental Evaluation 161
Nürn-
berger
Str.
Zuffenhausen
Feuerbach
Universität
Österfeld
Rohr
Goldberg
Hulb
Ehningen
Gärtringen
Nufringen
Oberaichen
Leinfelden
Echterdingen
Untertürkheim
Neckarpark
Mettingen
Oberesslingen
Zell
Altbach
Obertürkheim
Maubach
Nellmersbach
Winnenden
Schwaikheim
Neustadt-Hohenacker
Waiblingen
Fellbach
Sommer-
rain
Korntal
Ditzingen
Höfingen
Rutesheim
Renningen
Malmsheim
Rommelshausen
Stetten-Beinstein
Endersbach
Beutelsbach
Grunbach
Geradstetten
Winterbach
Weiler
Ludwigsburg
Kornwestheim
Tamm Asperg
Benningen (N)
Freiberg (N)
Favoritepark
Esslingen (N)
Leonberg
Nordbahnhof
Bad
Cannstatt
Plochingen
Wernau (N)
Wendlingen (N)
Ötlingen
Sindelfingen
Backnang
Erdmannhausen
Kirchberg (M)
Burgstall (M)
Marbach (N)
Stadtmitte
Feuersee
Maichingen
Maichingen Nord
Magstadt
Renningen Süd
Neuwirtshaus
(Porscheplatz)
Weilimdorf
Vaihingen
Schorndorf
Herrenberg
Bietigheim
-Bissingen
Weil der Stadt
Flughafen / Messe
Filderstadt
Kirchheim (T)
Haupt-
bahn-
hof
Böblingen
Schwabstraße
©
VVS 05.2016
Kundendialog DB Regio Baden-Württemberg: 0711 2092 7087 www.s-bahn-stuttgart.de
im Auftrag des
vvs
S-Bahn-Liniennetz
{5}
{4}
{2, 3}
{3}
{2}
{1}
{2, 3}
{2}
{1}
{1}
{6, 60}
{6}
{1, 2, 3, 4, 5, 6, 60}
{4, 5, 6, 60}
{1, 2, 3}
{60}
CR2
CR2
(1)
|Ω| = 1.74 · 10
8
{5}
{4}
{2, 3}
{3}
{2}
{1}
{2, 3}
{2}
{1}
{1}
{6, 60}
{6}
{1, 2, 3, 4, 5, 6, 60}
{4, 5, 6, 60}
{1, 2, 3}
{60}
UT7
(2)
|Ω| = 1.74 · 10
8
{1, 2, 3}
{5}
{4}
{2, 3}
{3}
{2}
{1}
{2, 3}
{2}
{1}
{1}
{6, 60}
{6}
{4, 5, 6, 60}
{60}
{1, 2, 3}
UT3
UT1
(3)
|Ω| = 5760
{5}
{4}
{3}
{2}
{2}
{6, 60}
{6}
{4, 5}
{60}
{2, 3}
{1}
UT1
UT1
UT6
(4)
|Ω| = 8
{5}
{4}
{3}
{2}
{6}
{60}
{1}
(5)
|Ω| = 1
Figure 4.41: Top: Ocial 2021 S-Bahn system map of Stuttgart. For uniform crossing
and separation weights, a full simplication by Algorithm 4.3 gives an optimal line-
ordering solution with 0 separations and 2 crossings. Bottom: Intermediate line graphs
after several iterations of Algorithm 4.3 (Lines S1 and S11 were considered a single line
S1, and nodes contracted by Pruning Rule 1 (Node Contraction) are not depicted for
brevity). The applied rules (CR = Cutting Rule, UT = Untangling Rule) are highlighted.
162 Chapter 4. Line Ordering Optimization
Table 4.3: Line graphs used in our experimental evaluation. Under |S| we give the
number of station nodes in the graph. Under |V | we give the numer of nodes (including
station nodes). Under |E| we give the number of edges. Under |L| we give the number
of lines. M is the maximum number of lines per edge. D is the maximum node degree.
|Ω| is the line-ordering search space size on the original line graph.
|S| |V | |E| |L| M D |Ω|
Freiburg (FR) 74 76 79 5 4 4 3×10
9
Dallas (DA) 108 117 118 7 4 4 6×10
20
Chicago (CG) 143 153 154 8 6 4 4×10
33
Sydney (SD) 175 198 202 9 6 3 9×10
58
Sydney (ST) 192 218 228 15 8 4 1×10
105
Turin (TU) 339 398 435 14 5 4 2×10
89
New York (NY) 456 517 548 26 9 5 2×10
269
version (for example, i-ILP optimizes MLNCM-W using our improved ILP formulation,
and i-ILP
∗
optimizes MLNCM-WS).
To ensure a realistic evaluation setup, we used manually adjusted crossing and sep-
arations weights which proved to produce line orderings that are very close to those in
maps by professional designers, as will be shown in Section 4.11.5. The weights were as
follows: At a node v, split-crossings at non-station nodes were weighted by 1 × deg(v),
split-crossings at station nodes were weighted by 3×deg(v), intra-path crossings at non-
station nodes were weighted by 4 × deg(v), intra-path crossings at station nodes where
weighted by 12 ×deg(v), separations at non-station nodes were weighted by 3 ×deg(v),
separations at station nodes were weighted by 9 ×deg(v). Additionally, we set a special
weightW higher than any of the weights above for crossings and separations at degree-2
stations.
Section 4.11.1 gives an overview of our main results. In Section 4.11.2 we compare
the results of our heuristic approaches described in Section 4.6. The performance of
our ILP on three dierent solvers (GLPK, COIN-OR CBC, and gurobi) will be evaluated
in Section 4.11.3. Section 4.11.4 will then evaluate the eect of the simplication rules
described in Section 4.9 on both the heuristic approaches, and the linear programs. The
full evaluation setup can be found online
1
.
4.11.1 Main Results
Our main results are summarized in Tables 4.4 and 4.5. The key takeaways are as follows:
1
https://github.com/ad-freiburg/loom-eval
4.11 Experimental Evaluation 163
Table 4.4: Main results for the running times of a selection of our heuristic and exact
(based on an ILP) methods on all test datasets. Under ‘graph‘ we give the simplica-
tion applied to the input graph: either none (raw), pruning & cutting (pr/cut) or full
simplication (f-simp).
heuristic exact
method GrLa
∗
Hill
∗
Hill
∗
Hill
∗
Ann
∗
b-ILP
∗
b-ILP
∗
i-ILP
∗
i-ILP
∗
i-ILP
∗
graph raw raw pr/cut f-simp f-simp raw f-simp raw pr/cut f-simp
FR 1ms 4ms 2ms 0.8ms 21ms 1.2s 4ms 21ms 18ms 4ms
DA 1ms 14ms 4ms 0.8ms 58ms 0.9s 54ms 31ms 15ms 6ms
CG 2ms 50ms 11ms 9ms 0.4s 2h 32.1s 0.7s 0.2s 0.3s
SD 3ms 0.1s 31ms 29ms 0.5s — 34m 1.1s 0.6s 0.6s
ST 5ms 0.5s 0.1s 21ms 0.6s — 48.4s 5.8s 1.2s 0.4s
TU 14ms 0.4s 0.1s 74ms 2.7s 30m 1m 1.7s 0.7s 0.5s
NY 25ms 3.4s 0.7s 86ms 1.0s — 8m 20.4s 2.1s 0.6s
Heuristic approaches are not good enough. While some heuristic approaches (in par-
ticular GrLa) give good results in terms of quality in the MLNCM-W setting, the
results were generally unsatisfactory for MLNCM-WS.
The improved ILP is much faster. Our improved ILP formulation (i-ILP) is several
orders of magnitude faster than the baseline formulation (b-ILP), in both the MLNCM-
W and MLNCM-WS setting. For many datasets, the baseline ILP could not be opti-
mized in under 6 hours.
Line graph simplication helps a lot. Full line graph simplication using the prun-
ing, cutting, and untangling rules described above greatly reduced the optimization
time. In particular, when compared to only applying the relative simple pruning
and cutting rules, the untangling rules provided an additional speed up. Line graph
simplication also had a positive eect on the quality of our approximation ap-
proaches.
Perfect optimization is possible in an interactive setting. With full line graph sim-
plication applied, we were able to solve all test datasets to optimality in under 600
milliseconds, which enables our method to be used in an interactive setting (for
example a map editor).
For more speed, combine greedy approach with hillclimbing. If an ILP is not an
option, the greedy lookahead method with hillclimbing is a viable alternative.
164 Chapter 4. Line Ordering Optimization
Table 4.5: Main results for the relative approximation error η =
θ
approx
θ
−1 of a selection
of our heuristic methods on all test datasets.
method Gr
∗
GrLa
∗
GrLa
∗
GrLa+
Hill
∗
GrLa+
Hill
∗
GrLa+
Ann
∗
GrLa+
Ann
∗
Hill
∗
Hill
∗
Ann
∗
Ann
∗
graph raw raw f-simp raw f-simp raw f-simp raw f-simp raw f-simp
FR 0.7 0.9 0.9 0.9 0.5 2.3 0.1 6.1 0.2 1.7 0.1
DA 1.7 1.0 1.0 1.0 1.0 23.3 0.3 44.2 2.2 25.5 0.7
CG 1.2 0.3 0.4 0.3 0.4 5.1 0.1 7.3 0.5 3.6 0.1
SD 2.3 0.9 1.7 0.9 1.3 6.0 1.7 10.8 3.2 6.1 1.2
ST 2.2 0.1 0.1 0.1 0.1 7.3 0.3 11.5 1.1 7.0 0.3
TU 3.2 1.0 1.7 0.7 0.4 4.0 0.8 7.0 1.4 3.8 0.6
NY 10.1 1.9 2.0 0.9 0.5 23.2 0.7 38.4 1.8 21.4 0.8
avg 3.0 0.9 0.9 0.7 0.6 10.2 0.6 17.9 1.5 9.9 0.6
4.11.2 Comparison of Heuristic Approaches
We evaluated the heuristic approaches described in Section 4.6 on all datasets and mea-
sured the running time, the nal objective value θ, and the relative approximation error η
when compared to the optimal solution obtained via our ILP. The relative approximation
error η is dened as follows:
η =
θ
approx
θ
− 1, (4.23)
where θ is the optimal objective value (a relative approximation error of 1 means that the
solution is twice as bad as the optimal solution). For each input line graph, we evaluated
both the full unmodied baseline graph, and the pruned and cut line graph (the eect
of additionally applying the untangling rules will be evaluated in Section 4.11.4). For
probabilistic approaches (Ann, GrLa+Ann) and hill climbing with random initialization
the results are the averages of 10 optimization runs. The results are given in Table 4.6.
For brevity, we only give the results for 3 datasets there. The average results over all
datasets are given in Table 4.7.
Solution Times
On the raw input graph, an exhaustive search (Exh) could not nd a solution in under
6 hours even on our smallest dataset (Freiburg). Using a carefully optimized implemen-
tation of the target function calculation method described in Section 4.6.1, we were able
to test around 30,000 line ordering solutions per second on average, but even with that
4.11 Experimental Evaluation 165
Table 4.6: Comparison of our approximate approaches on the raw input graph, and
the pruned & cut graph. We give the solution space size Ω, the time t, the number of
crossings ×, the number of separations ||, the target function value θ , and the relative
approximation error η.
On raw input graph On pruned & cut graph
|Ω| t × || θ η |Ω| t × || θ η
FR Exh — — — — — 0.1s 3.0 2.0 21.0 0.0
Gr 0.4ms 5.0 1.0 39.0 0.9 0.1ms 5.0 1.0 39.0 0.9
GrLa 0.4ms 3.0 2.0 21.0 0.0 0.1ms 3.0 2.0 21.0 0.0
+Hill 3×10
9
1ms 3.0 2.0 21.0 0.0 5×10
3
0.5ms 3.0 2.0 21.0 0.0
+Ann 0.2s 5.8 1.4 61.2 1.9 52ms 4.8 1.6 42.0 1.0
Hill 5ms 10.3 2.9 159.0 6.6 1ms 6.6 1.3 55.5 1.6
Ann 0.2s 6.0 2.1 63.9 2.0 52ms 3.6 1.4 29.1 0.4
Exh
∗
— — — — — 0.1s 5.0 0.0 39.0 0.0
Gr
∗
0.5ms 5.0 1.0 66.0 0.7 0.1ms 5.0 1.0 66.0 0.7
GrLa
∗
0.4ms 3.0 2.0 75.0 0.9 0.1ms 3.0 2.0 75.0 0.9
+Hill
∗
3×10
9
1.0ms 3.0 2.0 75.0 0.9 5×10
3
0.6ms 3.0 2.0 75.0 0.9
+Ann
∗
0.2s 7.3 1.5 130.2 2.3 42ms 4.8 1.4 81.9 1.1
Hill
∗
4ms 11.1 2.4 278.1 6.1 2ms 6.2 0.9 78.0 1.0
Ann
∗
0.1s 7.0 1.3 104.7 1.7 41ms 5.0 1.2 75.3 0.9
SD Exh — — — — — — — — — —
Gr 3ms 24.0 15.0 183.0 1.0 0.5ms 27.0 13.0 319.0 2.5
GrLa 3ms 12.0 1.0 183.0 1.0 0.6ms 17.0 5.0 225.0 1.5
+Hill 9×10
58
6ms 12.0 1.0 183.0 1.0 8×10
20
7ms 12.0 3.0 111.0 0.2
+Ann 2.7s 34.2 24.7 435.5 3.8 2.0s 18.7 10.4 124.7 0.4
Hill 0.1s 52.5 38.7 728.7 7.1 44ms 29.4 13.3 208.5 1.3
Ann 2.6s 38.7 25.6 448.4 4.0 1.8s 19.0 11.5 127.7 0.4
Exh
∗
— — — — — — — — — —
Gr
∗
3ms 24.0 15.0 354.0 2.3 0.5ms 27.0 13.0 526.0 3.9
GrLa
∗
3ms 12.0 1.0 210.0 0.9 0.6ms 17.0 5.0 288.0 1.7
+Hill
∗
9×10
58
6ms 12.0 1.0 210.0 0.9 8×10
20
6ms 18.0 5.0 249.0 1.3
+Ann
∗
1.3s 47.4 21.6 754.5 6.0 0.6s 26.7 7.8 261.2 1.4
Hill
∗
0.1s 60.8 29.4 1278.6 10.8 31ms 34.6 11.0 406.3 2.8
Ann
∗
1.3s 42.6 22.5 762.6 6.1 0.6s 27.0 9.2 264.1 1.4
NY Exh — — — — — — — — — —
Gr 22ms 76.0 32.0 1107.0 6.9 3ms 70.0 14.0 342.0 1.4
GrLa 25ms 51.0 13.0 294.0 1.1 3ms 50.0 15.0 309.0 1.2
+Hill 2×10
269
99ms 47.0 12.0 234.0 0.7 1×10
92
35ms 45.0 12.0 171.0 0.2
+Ann 11.7s 156.6 102.5 2431.0 16.2 7.1s 76.0 32.0 354.9 1.5
Hill 3.7s 214.6 144.6 4566.7 31.4 0.8s 104.1 53.0 603.9 3.3
Ann 12.1s 158.9 103.4 2526.6 16.9 7.4s 74.3 36.5 360.6 1.6
Exh
∗
— — — — — — — — — —
Gr
∗
22ms 76.0 32.0 2133.0 10.1 3ms 70.0 14.0 612.0 2.2
GrLa
∗
25ms 51.0 13.0 564.0 1.9 3ms 50.0 15.0 597.0 2.1
+Hill
∗
2×10
269
0.1s 44.0 10.0 357.0 0.9 1×10
92
56ms 45.0 13.0 351.0 0.8
+Ann
∗
6.8s 178.4 80.6 4655.1 23.2 3.5s 96.4 24.5 736.8 2.8
Hill
∗
3.4s 251.8 108.0 7557.9 38.4 0.7s 143.1 39.9 1344.9 6.0
Ann
∗
8.0s 181.2 77.3 4299.9 21.4 3.7s 96.1 25.1 722.4 2.8
166 Chapter 4. Line Ordering Optimization
Table 4.7: Average solution time (t) and relative approximation errors (η) of all our
heuristic approaches, over all datasets.
On baseline graph On pruned & cut graph
t η t η
Gr 7ms 2.3 0.9ms 1.3
GrLa 7ms 0.4 1.0ms 0.6
+Hill 25ms 0.4 14ms 0.1
+Ann 4.5s 6.9 2.8s 0.8
Hill 0.7s 13.1 0.2s 1.6
Ann 4.3s 6.5 2.8s 0.7
Gr
∗
7ms 3.0 0.8ms 1.8
GrLa
∗
7ms 0.9 1.0ms 1.1
+Hill
∗
23ms 0.7 13ms 0.7
+Ann
∗
2.6s 10.2 1.5s 1.5
Hill
∗
0.6s 17.9 0.1s 2.5
Ann
∗
2.6s 9.9 1.5s 1.4
throughput, it would take more than a day to explore the entire search space. On the
pruned & cut graph, however, we were able to nd optimal solutions using an exhaustive
search for Freiburg and Dallas in under 100 ms.
The greedy approaches (Gr, GrLa) were very fast, with an average solution time of 7
ms. On a randomized initial ordering solution, Hill converged to local optimum in under
0.6 second on average with line separations considered, which improved to 0.1 seconds
on the pruned and cut graph. Simulated annealing (Ann) was slower, with an average
solution time of 2.6 seconds on the raw, and 1.5 seconds on the pruned & cut graph,
both with line separations considered. We note that we did only little tweaking of the
annealing parameters. Faster cooling might lead to drastically improved solution times
for Ann.
If we combined the greedy lookahead (GrLa) approach with hill climbing and simu-
lated annealing, GrLa+Hill converged in under 30 ms on average for all settings (unfor-
tunately, this also meant that only little improvement was made, see the next section for
details). The solution times for GrLa+Ann remained roughly the same.
Quality
For the Freiburg dataset, the greedy approach with lookahead (GrLa) produced an op-
timal line ordering if no line separations were considered. For all other datasets, the
4.11 Experimental Evaluation 167
target function values were typically much worse than the optimal values in the raw
input graph. For example, on the raw graph of the New York subway dataset with line
separations considered, even the best-performing heuristic approach (GrLa+Hill) had a
relative approximation error of 0.9, meaning that the target function was nearly 2 times
higher than the optimal one. If GrLa was combined with hill climbing, the quality im-
proved only a little. If GrLa was combined with simulated annealing on the raw graph,
the quality greatly deteriorated. It seems likely that the simulated annealing quickly
leaves the solution provided by the initial GrLa approach because of the initial high
temperature and cannot nd its way back. If the graph was pruned and cut rst, this
eect was extenuated, but still measurable.
In general, pruning and cutting greatly improved the quality of the local search ap-
proaches using a random initial ordering, but had only little eect on the greedy ap-
proaches. This was to be expected, though: neither the pruned degree-2 nodes, nor the
cut edges with only one line play any role in the decision process for nding the relative
ordering between two lines in the GrLa approach.
On average, the greedy lookahead method combined with subsequent hillclimbing
(GrLa+Hill) performed best, achieving an average relative approximation error of only
0.2 in the MLNCM-W settings. If line separations were considered, however, the target
function values were, on average, nearly twice as high.
4.11.3 Comparison of Linear Programs
We evaluated all our testing datasets on three dierent ILP solvers: the GNU Linear Pro-
gramming Kit 4.65 (GLPK), COIN-OR CBC 2.10.3, and gurobi 9.1.2 (GLPK and COIN-OR
CBC are open source and publicly available, gurobi requires a license). For all solvers, we
used the default parameters (GLPK, gurobi) or reconstructed the default command line
parameters for the library (COIN-OR CBC). The evaluation results for our ILP variants
(b-ILP, b-ILP
∗
, i-ILP, i-ILP
∗
) on the raw input graph as well as on the pruned & cut graph
can be seen in Table 4.8. As for the heuristic approaches, the eects of the untangling
rules on the solution times will be evaluated separately in Section 4.11.4.
We observed a clear (and expected) performance ranking of the ILP solvers: gurobi
usually required the lowest solution times, closely followed by COIN-OR. GLPK followed
with some distance. This performance gap only emerged in larger ILPs, though, as the
various pre-solving techniques applied by COIN-OR and gurobi often only seem to pay
o for larger ILPs. In general, our baseline ILP formulation proved to be impractical on
the raw input graph. We were not able to optimize most of the datasets with any solver
in under 6 hours, except Freiburg and Dallas. Pruning & cutting did not help a lot here,
and even if it did, solution times on the best performing solver (gurobi) were still up to
several minutes.
In contrast, we were already able to optimize every dataset using the raw input graph
168 Chapter 4. Line Ordering Optimization
Table 4.8: ILP Dimensions (given as rows × cols) and solution times for all our ILP
variants on the raw input graph and on the pruned & cut input graph. If a graph had
multiple components, we optimized them separately, and the dimensions for the largest
component are given, but solution times are always cumulative. We only measured the
time to solve the ILP, not the ILP construction time or the time it took to extract the line
ordering from the ILP solution (which were both negligible) or the time required to set
up the solver environment.
On raw graph On pruned graph
rows×cols GLPK CBC GU rows×cols GLPK CBC GU × ||
FR b-ILP 3.0k ×268 13.5s 11.4s 0.4s 1.1k ×96 0.4s 3.4s 0.1s 3 2
b-ILP
∗
5.7k ×324 52.4s 46.8s 1.2s 2.0k ×114 1.0s 8.1s 0.2s 5 0
i-ILP 399 ×297 0.1s 22ms 8ms 144 ×105 10ms 13ms 8ms 3 2
i-ILP
∗
487 ×339 0.6s 0.1s 21ms 182 ×123 25ms 53ms 18ms 5 0
DA b-ILP 4.8k ×500 1m 39.5s 1.0s 1.4k ×161 1.4s 8.0s 0.2s 3 1
b-ILP
∗
8.9k ×638 2m 1m 0.9s 2.6k ×195 1.6s 7.5s 0.2s 3 0
i-ILP 799 ×572 0.3s 99ms 23ms 252 ×181 22ms 16ms 9ms 3 0
i-ILP
∗
991 ×662 1.8s 0.2s 31ms 317 ×211 0.1s 25ms 15ms 3 0
CG b-ILP 38.2k ×861 — — 48m 8.3k ×265 — 1h 26.5s 16 5
b-ILP
∗
72.8k ×1.1k — — 2h 15.4k ×311 — 1h 44.7s 21 0
i-ILP 1.4k ×982 33.1s 0.6s 51ms 393 ×284 0.1s 0.2s 13ms 16 4
i-ILP
∗
1.9k ×1.2k 2h 19.7s 0.7s 504 ×337 45.6s 4.1s 0.2s 21 0
SD b-ILP 97.2k ×1.4k — — 5h 50.1k ×537 — — 4m 13 5
b-ILP
∗
182.7k ×1.9k — — — 94.0k ×699 — — 1h 13 0
i-ILP 2.4k ×1.6k 2m 0.7s 99ms 873 ×598 9.2s 0.4s 45ms 13 5
i-ILP
∗
3.2k ×2.0k 1h 33.7s 1.1s 1.2k ×756 11m 5.7s 0.6s 13 0
ST b-ILP 206.6k ×2.4k — — — 44.2k ×928 — — 2h 39 14
b-ILP
∗
379.6k ×3.2k — — — 83.1k ×1.1k — — — — —
i-ILP 4.2k ×2.8k 9m 2.1s 0.1s 1.5k ×1.0k 20.7s 0.2s 33ms 39 9
i-ILP
∗
5.7k ×3.6k — 7m 5.8s 2.1k ×1.3k — 34.7s 1.2s 40 5
TU b-ILP 23.3k ×2.2k — — 8m 11.7k ×897 — — 33.4s 45 7
b-ILP
∗
43.0k ×2.8k — — 30m 21.7k ×1.1k — — 3m 50 2
i-ILP 3.5k ×2.5k 52m 0.8s 91ms 1.4k ×986 2m 0.4s 64ms 45 10
i-ILP
∗
4.5k ×3.0k — 27.3s 1.7s 1.8k ×1.2k — 12.2s 0.7s 50 2
NY b-ILP 248.0k ×6.0k — — — 95.8k ×2.3k — — — — —
b-ILP
∗
467.3k ×7.9k — — — 179.5k ×2.8k — — — — —
i-ILP 10.2k ×7.0k — 3.5s 0.2s 3.7k ×2.6k — 2.3s 100ms 43 17
i-ILP
∗
13.6k ×8.6k — 14m 20.4s 4.9k ×3.2k — 1m 2.1s 48 3
4.11 Experimental Evaluation 169
Table 4.9: Eects of full line graph simplication on line graph dimensions. |V | is the
number of nodes, |E| is the number of edges, M is the maximum number of lines per
edge, |Ω| is the search space size (sum of the search space sizes of the graph compo-
nents), C is the number of nontrivial (more than 2 nodes) graph components, C
1
is the
number of nontrivial graph components with a search space size of 1 (not requiring
further optimization).
Raw input graph Pruned & cut graph Fully simplied graph
|V | |E| M |Ω| C C
1
|V | |E| M |Ω| C C
1
|V | |E| M |Ω| C C
1
FR 76 79 4 3×10
9
1 0 36 30 4 5×10
3
4 0 19 15 4 2×10
2
1 0
DA 117 118 4 6×10
20
2 1 44 35 4 8×10
6
4 3 38 27 4 2×10
3
4 3
CG 153 154 6 4×10
33
1 0 27 26 6 5×10
9
2 1 24 22 6 1×10
9
2 1
SD 196 202 6 9×10
58
1 0 64 56 6 8×10
20
5 2 51 42 6 6×10
19
3 2
ST 218 228 8 1×10
105
1 0 84 79 8 2×10
37
6 2 69 58 6 8×10
12
2 0
TU 398 435 5 2×10
89
1 0 237 202 5 6×10
36
19 12 203 165 5 5×10
31
16 12
NY 517 548 9 2×10
269
2 1 173 176 9 1×10
92
9 1 119 96 6 4×10
36
3 1
med
red
1×10
38
1×10
39
with our improved ILP on an open-source solver (CBC), although it still took several
minutes on larger instances. However, an important takeaway is that in the MLNCM-
WS setting without any prior line graph simplication, even a highly sophisticated solver
like gurobi has solution times too long for interactive use (for example, the New York
network took over 20 seconds to optimize if line separations were considered).
Pruning & cutting already helped a lot and enabled us to solve every instance of
MLNCM-WS to optimality using our improved ILP in under 2.1 seconds (the best free
solver (CBC) still required over a minute on the largest dataset, New York).
4.11.4 Eects of Full Line Graph Simplication
We have seen above that the pruning & cutting rules already help a lot in two ways:
they sped up the solution times of both the approximate approach and the ILPs (some-
times dramatically), and often also improved the average quality of the approximate
approaches. In this section, we evaluate the eect of an additional application of the
untangling rules described in Section 4.9.4.
Table 4.9 gives an overview of the eect of full line graph simplication when com-
pared to only applying pruning & cutting. While pruning & cutting already had a
tremendous eect on the search space size (reducing it by nearly 200 orders of magni-
170 Chapter 4. Line Ordering Optimization
Table 4.10: Impact of full simplication on ILP sizes and solution times. We only mea-
sured the time to solve the ILP, not the ILP construction time or the time it took to extract
the line ordering from the ILP solution (which were both negligible) or the time required
to set up the solver environment.
Pruned graph Fully simplied graph
rows×cols GLPK CBC GU rows×cols
max
GLPK CBC GU
FR i-ILP 144 ×105 10ms 13ms 8ms 0 ×0 4ms 4ms 4ms
i-ILP
∗
182 ×123 25ms 53ms 18ms 0 ×0 4ms 4ms 4ms
DA i-ILP 252 ×181 22ms 16ms 9ms 123 ×90 7ms 10ms 7ms
i-ILP
∗
317 ×211 0.1s 25ms 15ms 149 ×102 13ms 13ms 6ms
CG i-ILP 393 ×284 0.1s 0.2s 13ms 371 ×267 0.3s 0.1s 15ms
i-ILP
∗
504 ×337 45.6s 4.1s 0.2s 482 ×320 27.9s 4.0s 0.3s
SD i-ILP 873 ×598 9.2s 0.4s 45ms 837 ×572 5.4s 0.2s 42ms
i-ILP
∗
1.2k ×756 11m 5.7s 0.6s 1.2k ×727 25m 10.2s 0.6s
ST i-ILP 1.5k ×1.0k 20.7s 0.2s 33ms 497 ×349 0.8s 0.2s 24ms
i-ILP
∗
2.1k ×1.3k — 34.7s 1.2s 678 ×435 1m 13.5s 0.4s
TU i-ILP 1.4k ×986 2m 0.4s 64ms 1.2k ×854 51.4s 0.3s 45ms
i-ILP
∗
1.8k ×1.2k — 12.2s 0.7s 1.6k ×1.0k — 10.5s 0.5s
NY i-ILP 3.7k ×2.6k — 2.3s 100ms 1.4k ×990 11.4s 0.2s 50ms
i-ILP
∗
4.9k ×3.2k — 1m 2.1s 1.9k ×1.2k — 8.6s 0.6s
tude for the New York dataset), the additional application of the untangling rules again
gave a reduction of up to 57 orders of magnitude for the largest dataset, with a median
search space size reduction of 39 orders of magnitude. While all input test datasets con-
sisted of only a single nontrivial graph component with a search space size greater than
1, the pruning and cutting rules increased the number of nontrivial components to up
to 19 (for Turin). The untangling rules then decreased the number of components again.
For example, the Stuttgart light rail network had 6 nontrivial components after pruning
& cutting, but only 2 after the additional application of the untangling rules. This is
because the full line graph simplication completely untangled some components left
by the pruning & cutting rules into trivial components consisting of only a single edge,
an eect already demonstrated in the example of the simpler Stuttgart S-Bahn network
given in Figure 4.41.
The eect of full line graph simplication on the heuristic approximation approaches
4.11 Experimental Evaluation 171
Table 4.11: Impact of full simplication on selected baseline heuristic solution times
and objective function values.
Pruned graph Fully simplied graph
t × || θ η t × || θ η
FR Exh
∗
0.1s 5.0 0.0 39.0 0.0 4ms 5.0 0.0 39.0 0.0
GrLa
∗
0.1ms 3.0 2.0 75.0 0.9 < 1ms 3.0 2.0 75.0 0.9
+Hill
∗
0.6ms 3.0 2.0 75.0 0.9 0.5ms 3.0 1.0 60.0 0.5
+Ann
∗
42ms 4.8 1.4 81.9 1.1 21ms 4.8 0.2 43.8 0.1
Hill
∗
2ms 6.2 0.9 78.0 1.0 0.8ms 4.2 0.4 48.0 0.2
Ann
∗
41ms 5.0 1.2 75.3 0.9 21ms 5.0 0.1 41.7 0.1
SD Exh
∗
— — — — — — — — — —
GrLa
∗
0.6ms 17.0 5.0 288.0 1.7 0.5ms 17.0 5.0 288.0 1.7
+Hill
∗
6ms 18.0 5.0 249.0 1.3 6ms 18.0 5.0 249.0 1.3
+Ann
∗
0.6s 26.7 7.8 261.2 1.4 0.4s 29.1 9.4 290.6 1.7
Hill
∗
31ms 34.6 11.0 406.3 2.8 29ms 34.6 12.4 451.5 3.2
Ann
∗
0.6s 27.0 9.2 264.1 1.4 0.5s 23.6 9.7 242.1 1.2
NY Exh
∗
— — — — — — — — — —
GrLa
∗
3ms 50.0 15.0 597.0 2.1 1ms 50.0 13.0 570.0 2.0
+Hill
∗
56ms 45.0 13.0 351.0 0.8 23ms 49.0 9.0 282.0 0.5
+Ann
∗
3.5s 96.4 24.5 736.8 2.8 1.0s 59.8 9.5 329.7 0.7
Hill
∗
0.7s 143.1 39.9 1344.9 6.0 86ms 82.0 18.2 538.0 1.8
Ann
∗
3.7s 96.1 25.1 722.4 2.8 1.0s 61.2 9.7 342.4 0.8
avg GrLa
∗
1.1 1.1
+Hill
∗
0.7 0.6
+Ann
∗
1.5 0.6
Hill
∗
2.5 1.5
Ann
∗
1.4 0.6
172 Chapter 4. Line Ordering Optimization
Figure 4.42: Left: Optimal line ordering for the Chicago Loop using our MLNCM-WS
formulation. Right: Line ordering in the ocial map. A single line swap of the blue
and brown line between the lower right and the upper left intersection transforms our
line ordering into the ocial line ordering. This was the only line swap required on the
entire map.
was twofold: First, we measured an additional speedup across all techniques. Second,
full simplication resulted in an additional quality boost for nearly all our methods. With
full simplication, the average approximation error for our informed greedy search with
lookahead combined with subsequent local search (hill climbing) slightly went down to
0.6 from 0.7 on the pruned & cut input graph. For other approaches, the eect was more
pronounced: using simulated annealing on the fully simplied graph gave a relative
average approximation error of 0.6, compared to 1.4 on the pruned & cut graph, and 9.9
on the raw input graph.
4.11.5 Comparison to Manually Designed Maps
An evaluation of the esthetic quality of our line orderings is dicult as there are no
established quality metrics (except for the number of crossings). Nevertheless, we com-
pared the optimal line orderings of the Freiburg, Dallas, Chicago, and Stuttgart networks
to existing ocial maps, created by professional map designers. We compared them to
our MLNCM-WS approach (with line crossings and line separations penalized), using the
manually tuned crossing and separation weights mentioned above. As the ocial maps
and their line orderings are not available in machine-readable form, we hand-counted
the crossings and separations from published PDF maps and calculated the value of the
4.12 Conclusions and Future Work 173
Ocial map MLNCM-WS
× || θ × || θ T
FR 7 1 132 6 0 48 2
DA 3 1 27 3 0 9 1
CG 26 0 80 27 0 80 1
ST 65 5 264 64 2 156 4
Table 4.12: Comparison of the line or-
derings in professionally designed (o-
cial) maps, and the line orderings pro-
duced by our approach. Under T we
give the number of line swaps neces-
sary to transform one ordering into the
other (we only counted swaps between
the same lines along a path once).
target function by hand. For Dallas and Chicago, a single line swap was enough to trans-
form our ordering into the ocial ordering (see Figure 4.42 for an example). Even the
highly complex Stuttgart network only required 4 line swaps.
4.12 Conclusions and Future Work
This chapter discussed the problem of nding optimal line orderings in metro maps,
a crucial task for ensuring both readability and esthetic quality of the nal map. We
gave an alternative formulation of the classic Metro Line Crossing Minimization (MLCM)
problem, namely the Metro Line Node Crossing Minimization (MLNCM) problem, which
only allows crossings at nodes, not on edges. To produce maps that are more estheti-
cally pleasing, and to give map designers some control over the crossing placements, we
introduced 3 variants of MLNCM: a weighted version (MLNCM-W), a version in which
line separations are penalized (MLNCM-S), and a version in which both crossings and
separations are weighted (MLNCM-WS). We have shown that MLNCM, MLNCM-W, and
MLNCM-WS are NP-hard on general graphs, while the complexity of MLNCM-S is an
open problem. We gave a linear time algorithm for MLNCM on graphs in which all
lines follow simple paths and terminate at nodes of degree 1. Even on these instances,
MLNCM-W remains NP-hard. We gave several heuristic approximation algorithms for
the most general problem (MLNCM-WS), and gave a baseline ILP to solve the problem to
optimality. As the baseline ILP required impractically high solution times, we formulated
an improved variant.
To further speed up the optimization, we described several pruning, cutting, and
untangling rules which may be applied to the line graph before optimization. Their
benet lies in the fact that they may not only reduce the search space size by many
orders of magnitude, but also typically split the line graph into connected components,
enabling parallel optimization. We proved that none of these simplication rules aect
the optimality of the nal line ordering and showed their eectiveness in our evaluation.
We described instances for which the simplication rules already nd an optimal line
ordering, as they decompose the input graph into components with search space sizes
174 Chapter 4. Line Ordering Optimization
𝑣
{𝐵}
{𝐴,𝐶}
{𝐴, 𝐵, 𝐶}
𝑔
𝑓
𝑒
{𝐴, 𝐵}
{𝐶}
𝑖
0
ℎ
0
𝑢
{𝐴, 𝐵}
{𝐶}
ℎ
𝑖
{𝐴, 𝐵}
{𝐶}
𝑒
0
𝑒
00
𝑢
0
𝑢
00
𝑣
{𝐵}
{𝐴,𝐶}
𝑔
𝑓
𝑖
0
ℎ
0
𝑒
0
𝑒
00
𝑢
0
𝑔
𝑓
𝑢
00
𝑣
𝐴
𝐵
𝐶
𝑖
ℎ
𝑒
𝑔
𝑓
𝑣
𝐴
𝐵
𝐶
𝑢
C: 1 S: 0
C: 1 S: 1
Figure 4.43: A generic ripping rule like described in this section only works if line
separations are not considered. In this example, applying the ripping rule to node u may
later hide a separation of lines A and C at v, resulting in a non-optimal line ordering.
of 1. Finally, we showed that with carefully chosen crossing and separation weights,
the line orderings found by our approach closely match those of professionally designed
maps.
We see several directions for future work. First, we do not claim that the line graph
simplication rules given in Section 4.9 are exhaustive. During the development of our
prototype, several ideas for additional untangling rules emerged: “Ripping” casts the
intuition behind our informed greedy search with lookahead from Section 4.6.4 into an
explicit untangling rule. Consider Figure 4.43: if the lines on an edge e = {u,v} branch at
u into n minor legs, split u into n nodes like in Untangling Rule 4 or 5 if all crossings and
separations may occur at equal or cheaper cost at v. The only dierence to Untangling
Rule 5 is that we do not require the minor leg mapping A. In this conguration, however,
a key property used in the correctness proofs of the previous untangling rules is lost:
there might now be separations between two lines from dierent minor legs at v, and
these separations will be hidden if u is split. Still, such a ripping rule may be applied if
line separations are not considered, or if line separation costs at v are 0.
The untangling rules described in Section 4.9.4 were built on the concepts of minor
and major legs. Each leg was a distinct edge. We may generalize this into an abstract
variant we call induced minor leg. The main dierence is that an induced minor leg may
consist of multiple edges. Figure 4.44 gives an example for an induced full X: although
there is no opportunity for any of the untangling rules dened above, sets of edges may
4.12 Conclusions and Future Work 175
L
1
= {A, B}
{B}
L
1
= {A, B}
e
i
j
i
0
{A}
f
L
1
= {A, B}
L
1
= {A, B}
{B}
{A}
e
0
f
0
k
L
2
= {C, D, E}
{E}
{C, D}
j
0
k
0
L
2
= {C, D}
{E}
{C, D}
д
h
{D}
{C, E}
v
L
2
= {C, D, E}
д
h
{D}
{C, E}
L
2
= {C, D, E}
v
v
0
Figure 4.44: Application of an induced untangling rule on a node v. Although no ad-
jacent edge fullls the requirements for a leg in Untangling Rule 1, the pair of edge sets
{e, f } and {i} does. Generally speaking, induced untangling rules search for a grouping
of adjacent edges in such a way that group sets may act as minor or major legs in the
untangling rules.
satisfy the requirements for a leg. For instance, edge i and the set of edges {e, f } share
the exact same lines and may therefore be detached from v
0
using a similar argument
as in the correctness proof for Untangling Rule 1. Such an induced version can be given
for all our untangling rules and may be able to further break down the line graph by
creating novel simplication opportunities.
As mentioned above, the crossing and separation weights used in our evaluation are
based on intuition and manual ne-tuning. A user study (for example, A/B testing) could
help to conclusively determine the weights which are both most esthetically pleasing and
informative.
With full line graph simplication using our pruning, cutting, and untangling rules,
we were able to optimize MLNCM-WS instances with realistic crossing and separations
weights in under 600 milliseconds for all testing datasets, enabling the use of our meth-
ods in an interactive setting. For settings where this is still not fast enough, a combina-
tion of an informed greedy search and subsequent hillclimbing on the fully simplied
line graph was shown to be a practical and fast alternative of often sucient quality.
Chapter 5
Schematic Transit Maps
So far, we have only considered geographically accurate transit maps in which lines
follow their real-world geographic course as close as possible. While these maps are es-
pecially well suited to be laid over existing maps or satellite imagery, the generation of
schematic transit maps is also of high practical interest. Transit maps which are posted
in stations or in vehicles, or are distributed in print usually show a high degree of simpli-
cation to ensure legibility. The de facto standard for schematized transit maps follows
the classic octilinear layout already mentioned in Chapter 1. The problem of gener-
ating such octilinear transit maps automatically has been studied extensively over the
last decades. In recent years, a number of maps following alternative layouts have been
published both by transit agencies and map enthusiasts [110]. This has led to an interest
in the automatic generation of transit maps which are not octilinear, but for example
orthoradial [13, 113] or curvilinear [58, 48]. Although this chapter will begin with a de-
scription of an approach to render octilinear maps, we will later extend our method to
other layouts. We consider the following problem:
Problem 5 (Line Graph Schematization). Given a line graph G = (V , E, L, L), nd a
schematic drawing D
G
= (P, C) of G such that P(v) ∈ R
2
assigns a position to every node
v ∈ V and C(e) = (q
0
, q
1
, . . . , q
n−1
), q
i
∈ R
2
assigns a piece-wise linear curve to every edge
e ∈ E. D
G
should resemble a classic schematic transit map.
The following section will make it clear how the schematic transit map resemblance
is formalized. After some preliminaries and a discussion of related work, we will rst de-
scribe and evaluate an approach to generate schematic transit map drawings in a classic
octilinear fashion by nding a set of optimal interconnected shortest paths on a specially
crafted octilinear base grid graph. We will then continue to extend our approach to other
base grids, both to further speed up the original approach and to render maps which are
not octilinear. The methods presented in this chapter have been previously published in
[21] and [22].
177
178 Chapter 5. Schematic Transit Maps
(1) (2) (3) (4)
Figure 5.1: The degree-2 heuristic typically applied to transit networks before schema-
tization. Degree-2 nodes in the initial input line graph (1) are contracted (2), the resulting
line graph is schematized (3), and contracted nodes are re-inserted equidistantly (4).
5.1 Preliminaries
To formally dene a drawing as a schematic transit map, a set of hard and soft constraints
is often used. The soft constraints constitute the objective function, and the hard con-
straints dene the set of feasible solutions. While these constraints have been originally
dened for octilinear embeddings [117, 118], they also provide a solid foundation for
other schematization types. As hard constraints, we thus consider the following:
H1 Schematic Representation. Each edge curve C(e) must follow the schematization
layout dened for the drawing type (e.g. octilinear, orthoradial, ...).
H2 Topology Preservation. The topology of the original input line graph must be pre-
served. In particular, this means that no crossings between edges must be intro-
duced, non-incident edges must not share common points and circular edge order-
ings around nodes must be preserved.
H3 Map Density. Distances between stations and curve anchor points must be higher
than a threshold D.
As soft constraints, we consider the following:
S1 Edge Monotony. Edge bends should be minimized and obtuse angles should be pre-
ferred.
S2 Edge Length. The total length of edges should be minimized.
S3 Geographical Accuracy. The station positions should reect the original geograph-
ical positions.
Our running assumption in this chapter is that these soft and hard constraints are
sucient to model the esthetic quality and readability of a map.
5.1.1 Degree-2 Heuristic
Soft constraint S3 (Geographical Accuracy) is often only applied to nodes with a degree
deg(u) , 2 (intersection nodes with deg(u) > 2, and terminus nodes with deg(u) = 1).
Line graph nodes of degree 2 are then contracted before schematization. The contracted
5.1 Preliminaries 179
e
8
e
9
e
10
e
1
e
4
e
7
e
2
e
3
e
5
e
6
v
→
e
1
e
4
e
7
v
0
f
e
0
8
e
0
9
e
0
10
e
2
e
3
e
5
e
6
v
Figure 5.2: Left: Node of an input line graph G with deg(v) > 8. Right: To be able to
draw G in an octilinear fashion, we add an additional non-station node v
0
, inect edges
e
8
, e
9
, e
10
to v
0
, and connect v and v
0
by an additional edge f holding the combined lines.
nodes are later re-inserted equidistantly onto the schematic drawing. This greatly re-
duces the complexity of the problem and has been found to improve the overall map
appearance [118]. We call this the degree-2 heuristic. Figure 5.1 gives an example.
5.1.2 Station Labels
As station labels are an essential part of schematic transit maps, labeling is often dis-
cussed together with map schematization. Two general approaches exist: A posteriori
labeling adds station labels after the schematic map has been generated. This amounts
to basically the same problem as labeling geographically accurate transit maps.
As station labels might require explicit free space around stations or inuence the
optimal layout, integrated labeling optimizes both the labeling and the schematic map
together. The labels act as rectangular obstacles for the stations and lines.
Labeling was not the focus of this work. A simple baseline approach to a posteriori
labeling is described in Chapter 6.
5.1.3 Extending the Maximum Input Node Degree
An input line graph G only allows for an octilinear representation if its maximum node
degree is ≤ 8. While real-world public transit networks typically satisfy this constraint,
this is not guaranteed. In the course of this chapter we will also introduce several alter-
native layouts which require a maximum input node degree of 6 or even 4.
To still allow for schematic drawings of input graphs whose maximum node degrees
are greater, we use the following approach: Given a node v with deg(v) > D and incident
edges e
1
. . . e
deg(v)
, as depicted in Figure 5.2. We rst add a new non-station node v
0
to G. Afterwards, for edges e
D+1
. . . e
deg(v)
, we substitute v for v
0
. An additional edge
180 Chapter 5. Schematic Transit Maps
f = {v
0
,v} is then added which holds all lines on the previously inected edges, that is
L(f ) =
e
D+1
...e
deg(v )
L(e). If we now still have deg(v
0
) > D, we repeat to process for v
0
.
Figure 5.2 depicts v
0
with a dierent position than v
0
. In practice we will move v
0
to the
same position as v to avoid distorting soft constraint S3.
5.2 Related Work
Our work is closely related to previous work on automatic generation of schematic tran-
sit maps (often called metro maps in the literature), and network schematization in gen-
eral. We use the terms metro map and schematic transit map interchangeably from now
on. There is also work on the user perception of dierent layout styles. For an extensive
overview and a detailed taxonomization of existing work we refer to a recent survey
by Wu et al. [151]. Additionally, there are several classic algorithmic problems that are
related to our method of nding metro map drawings on an auxiliary base grid.
5.2.1 Generation of Schematic Maps
A substantial body of work is concerned with the generation of schematic maps in gen-
eral, often for road network schematization. While these works are all applicable to the
problem of generating schematic transit maps (some of them explicitly mention this use
case), they often ignore specic aspects of schematic transit maps and typically come
without any experimental results on real-world transit networks.
Already in 1988, Elroi [54, 55] suggested three steps for the schematization of road
networks: (1) line simplication, (2) line re-orientation along a grid, and (3) enlarge-
ment of high density areas. Neyer [111] considered the problem of simplifying a single
polyline P into a schematic polyline Q consisting of segments following a predened set
of orientations (e.g. diagonal, vertical, or horizontal). Q should consist of the minimal
number of segments such that the Fréchet distance between P and Q is within some ϵ.
Avalar and Müller [12] described an approach to octilinearize road maps by specify-
ing a set of constraints per node (minimum distance to other network nodes, octilinear-
ity of adjacent edges and topology preservation). The network was rst simplied using
the Douglas-Peucker algorithm. The remaining nodes were then visited iteratively and
moved to a nearby position which did not violate any constraints (or left untouched if
this was already the case). Although the experiments were done with road networks,
the applicability of the approach to transit maps was explicitly noted (but not tested).
Final maps were only optimal in the sense that as many nodes as possible did fulll the
constraints, and users had to manually abort the iterative process as soon as they were
pleased with the current result. This approach was extended by Ware et al. [144] who
introduced a cost function to measure the quality of a map and additionally considered
5.2 Related Work 181
Avelar and Müller, 2000 [12] Cabello et al., 2001 [33] Ware et al., 2006 [144]
Li and Dong, 2010 [101]
Figure 5.3: Example results of previous work on schematic maps, taken from their pub-
lications. Avelar and Müller, Ware et al., and Li and Dong considered the schematization
of road networks (the examples all schematize dierent networks). Cabello et al. studied
mathematical properties of map schematization and used a toy example.
node displacement and edge rotation dierence. Simulated annealing was then used for
optimization. This approach was later rened to use gradient descent for optimization
by Anand et al. in [9].
Cabello et al. [33] investigated mathematical properties of drawing octilinear road or
railroad maps with at most three segments per edge, while preserving the topology of the
input network and maintaining a minimum distance between segments. An O(n log
3
n)
algorithm was described which returned such a schematization if one exists. As in [12],
no optimization took place - the returned solution was only guaranteed to be feasible.
In [101], a method for road network schematization based on the concept of strokes
(simple paths in the road network) was presented. Several methods to decompose the
road network into a set of strokes were presented. The strokes were then simplied
using the Douglas-Peucker algorithm and the segments of the simplied strokes were
then aligned along an octilinear grid. Afterwards, intersection and terminus nodes were
re-projected onto the strokes. An enlargement of high-density areas was later added as
a preprocessing step in [137].
Figure 5.3 gives some examples for the approaches described above.
5.2.2 Generation of Schematic Transit Maps
The specic problem of drawing public transit networks in a way resembling a schematic
transit map was rst introduced by Hong et al. in [84] and [85]. A set of esthetic crite-
ria was dened and several variations of the classic spring layout algorithm were used
to produce schematic graph embeddings resembling a metro map. Regarding topology
preservation, it was only required that the resulting drawing should not introduce any
edge crossings. Because the geography of the input stations was also ignored, the result-
ing maps could dier greatly from the topology of the original network. As a prepro-
182 Chapter 5. Schematic Transit Maps
Hong et al., 2004 [84]
Merrick and Gudmundsson,
2006 [106]
Stott et al., 2011 [134]
Nöllenburg and Wol, 2011
[118]
Wang and Chi., 2011 [142] Chivers and Rodgers, 2014 [38] Wang and Peng, 2016 [143]
Bast et al., 2020 [21]
Figure 5.4: Example results of previous work on octilinear transit maps, taken from
their publications.
cessing step, the degree-2 heuristic was introduced (see Section 5.1.1). Station labeling
was added a posteriori. Although octilinearity was identied as being a key feature of
esthetically pleasing metro maps, the resulting drawings usually did not fulll this.
Merrick and Gudmundsson [106] built on the work of Neyer [111] and proposed
a polyline simplication method specically designed for metro map generation. The
Fréchet distance used in [111] was found to produce unnaturally looking outliers in
some cases. They employed the Hausdor distance instead. To extend this line simpli-
cation method to networks, an ordering on the edges of the input network was de-
termined. The corresponding polylines where then simplied in this order. As soon as
an edge was simplied this way, the position of its adjacent network nodes was xed.
The schematization of subsequent edges then respected these xed nodes. The method
was tested on multiple real-world public transit networks. As the local simplication
did not consider other already simplied input edges, their approach did not guarantee
topological correctness.
Stott and Rodgers presented a more rened set of esthetic criteria in [133], thus
achieving greatly improved results. They applied local-search techniques to nd an op-
timal metro-map embedding by iteratively moving input nodes on a grid, selecting the
most promising new position candidate. In [134], Stott et al. rened this local search by
three clustering techniques which grouped nodes that were then moved together (thus
5.2 Related Work 183
enabling the local search to arrive at neighborhoods behind a local minimum). Labels
were optimized together with the layout. As the nal drawing was a node embedding
(and edges therefore straight lines between nodes on a grid), the main problem of their
approach was that the resulting embeddings were not guaranteed to be octilinear. Our
method may be understood as an extension of the method by Stott and Rodgers. The
major dierence is that in our case, edges in the nal schematic drawing are shortest
paths on an octilinear grid, instead of straight lines. This not only guarantees octilinear
results, but has other benets as well (for example, obstacle avoidance).
Nöllenburg and Wol described a novel approach in [117]. There, they introduced
the hard and soft constraints given in Section 5.1. The goal was to nd a node embed-
ding of a given line graph which preserved the topology, respected the original station
positions, and consisted of edges drawn as horizontal, vertical, or diagonal lines. Finding
such an embedding was shown to be NP-complete by a reduction from Planar 3-Sat
in [115]. A mixed-integer linear program (MLP) was described which optimized an ob-
jective function built from the soft constraints, subject to the hard constraints. In [118],
this MLP was extended to guarantee enough space around station markers for labels.
While the resulting maps were esthetically pleasing, it took considerable time to solve
the MLPs to optimality (see Table 5.1). Milea et al. [107] added constraints to the original
MLP formulation to ensure that the schematic map preserves shortest paths in the input
graph.
Wang and Chi [142] described a schematization approach in which the goal was to
dynamically focus on routes through the network to ensure readability on mobile devices
with small screens. The route focus is optional, though, and the method can be used to
generate classic metro maps of high quality. They rst compute a curvilinear map which
is then rened into octilinearity. The objective function was formulated in terms of
multiple energy terms and optimized in a least-squares sense. Reported schematization
times were very good, with typical maps being schematized in under 1 second. Labeling
was done a posteriori. This was the rst work to achieve a performance which allowed
for interactive use. It was later rened into a fully interactive editor which allowed map
editors to drag stations to specied positions [143].
In [153], Wu et al. built on the MLP by Nöllenburg and Wol to create maps where a
single route is highlighted as a straight horizontal line. They then optimized the place-
ment of large pictorial station labels, placed at the top or bottom of the map and con-
nected to stations via a curve. In [152], the labels were placed inside the map. This was
done in three steps: rst, an unlabeled layout was produced by an MLP, which was then
enlarged until there was enough space to hold the pictorial labels. Afterwards, the lay-
out was tightened again to remove unnecessary white space. Chivers and Rodgers [38]
reconsidered the force-based directed approach by Hong et al. [84]. They combined a
classic spring embedder to balance edge lengths with magnetic forces to snap the edges
into octilinear directions. To avoid conicting forces, they applied an iterative approach:
184 Chapter 5. Schematic Transit Maps
Publication Network t
Hong et al. [84] Sydney 2.3 s
Merrick and Gudmundsson [106] Sydney 0.3 s
Stott et al. [134] Sydney 7.2 s
Nöllenburg and Wol [118] Sydney 23 m
1
Wang and Chi [142] Sydney 0.8 s
Chivers and Rodgers [38] Sydney 0.6 s
Wang and Peng [143] Berlin 0.2 s
Bast et al. [22] Sydney 0.4 s
Table 5.1: Reported octilin-
earization times (all without
labeling) from previous work.
Wang and Peng also evaluated
on Sydney, but without reporting
the octilinearization time.
1
Suboptimal ILP solution, optimality gap of 16.4% after 12 h
initially, the magnetic forces are given only little weight to produce a balanced, but not
octilinear map. They then gradually shift the weight from the spring forces to the mag-
netic forces with each iteration to arrive at an octilinear result. To further polish the
maps, a nal post-processing phase is added to straighten peripheral edges. Van Dijk
and Lutz [49] described a fast approach to draw a graph such that a given uniform edge
length is realized, while the original edge directions should be preserved as good as pos-
sible. They formulated a set of constraints (preservation of input node positions and
edge length discrepancy) and optimized them in a least-squares sense. An extension of
their formulation also tries to optimize for octilinearity. While their approach is very
fast (with a typical runtime of under 10 ms), the resulting maps only approximate octi-
linearity and do not guarantee preservation of topology.
Earlier version of our own approach were described in [21] and [22]. Labeling was
done a posteriori. The main dierence between our approach and the methods described
above is that previous work was interested in octilinear node embeddings. Hence, each
segment between stations was required to be a straight line (horizontal, vertical or diag-
onal) in the nal drawing, with no edge bends (or the number of bends was xed to 2, as
in [33]). In contrast, our work tries to nd octilinear drawings where each segment can
have an arbitrary number of edge bends. This is motivated by the observation that real-
world schematic transit maps are usually not octilinear node embeddings, but octilinear
drawings. Our method allows for more exible layouts. In particular, it is able to route
around obstacles like rivers or lakes, or to approximate the real geographical course. We
note that Nöllenburg and Wol [117] already observed that after the degree-2 heuris-
tic was applied and an embedding for the terminus and intersection stations had been
found, the nal maps had long, unnatural looking straight lines between intersection
and terminus stations. Stations were also sometimes heavily displaced to arrive at an
embedding. Their solution was to re-add a number of explicit bend nodes after applying
the degree-2 heuristic and found that 2 bend nodes were enough to produce pleasing
results. A similar technique might be used to allow line bends between stations. How-
5.2 Related Work 185
Fink et al., 2012 [58]
van Dijk et al., 2014 [48]
Nickel et al., 2019 [112] Niedermann and Rutter, 2020 [113]
Figure 5.5: Example results of previous work on non-octilinear schematic maps, taken
from their publications.
ever, the number of required bend nodes is not known beforehand. Additionally, the
added bend nodes might unnecessarily elongate line segments because of the minimum
segment length required by all of the approaches described above.
5.2.3 Generation of Non-Octilinear Schematic Transit Maps
Several works investigate the problem of drawing non-octilinear schematic transit maps.
Fink et al. [58] used a force-based approach to move the control points of cubic Bèzier
curves to arrive at curvilinear layouts. They additionally applied checks to ensure that
the input topology was maintained. In particular, they strived for a layout where indi-
vidual transit lines are always drawn as continuous curves, even when they pass through
intersections. In [48], van Dijk et al. presented an approach where the network is rst
decomposed into a set of strokes which are then drawn as circular arcs. Nickel et al.
[112] extended the MIP by Nöllenburg and Wol [118] to be able to render k-linear de-
signs (following angles which are multiples of 360/2k). For k = 4, this is equivalent to
rendering octilinear maps.
Since Maxwell Roberts published a map of the London underground network mostly
composed of concentric circles (the usability of this layout was later investigated by
Newton and Roberts [110]) which received large media attention in Great Britain, or-
thoradial metro maps have found some research interest. Barth et al. noticed that an
orthoradial drawing may be interpreted as a rolled-up orthogonal drawing [13]. Nie-
dermann and Rutter [113] later formulated an ILP to render orthoradial drawings with
a minimum number of edge bends and tested their approach on transit maps. Their
method did not consider labeling. We will demonstrate that our own approach can also
be used to produce orthoradial drawings in Section 5.10. It has to be noted, though, that
manually designed concentric circle maps are usually not strictly orthoradial (see for
example the maps given in [110]). We try to acknowledge this in Section 5.10 where we
produce pseudo-orthoradial maps.
186 Chapter 5. Schematic Transit Maps
5.2.4 General Algorithmic Work
A number of classic algorithmic problems are related to drawing (octilinear) transit
maps. The rectilinear Steiner minimum tree (RSMT) problem asks to connect a set
S = {s
1
, . . . , s
n
} of points in the plane by a network of vertical and horizontal line seg-
ments. The resulting graph may contain additional “bend” nodes where a horizontal
segment changes to a vertical segment, or vice versa. The optimal such graph will al-
ways be a tree and has a striking resemblance to rectilinear transit maps. The main
dierence is that input points cannot be moved, and that connections are not part of
the input. RSMT is NP-complete [66]. Maurice Hanan introduced the Hanan grid H(S),
which is built from a set S of points on the plane by drawing vertical and horizontal
lines through each point in S, and adding nodes to their intersections [79]. He showed
that for a given set S, H(S) contains an RSMT for S. We will extend this concept to an
octilinear Hanan grid to reduce the search space size for octilinear transit maps.
Our approach to nd schematic drawings by routing input line graph edges through
a special base grid bears some resemblance to the k-node-disjoint path problem (NDP).
Given a graph G and k node pairs (s
1
, t
1
), . . . , (s
k
, t
k
), NDP asks for a set of k node-disjoint
paths p
1
, . . . , p
k
, where path p
i
connects node pair (s
i
, t
i
) (path start and end nodes do
not have to be disjoint). Its NP-completeness was shown by Karp in 1975 [91]. For xed
k, a number of polynomial algorithms exists. Robertson and Seymour showed that, at
least theoretically, there is a polynomial-time algorithm for any xed k in the undi-
rected case [126] (for directed graphs, the problem remains NP-complete for any xed
k > 2). The general case (were k is part of the input) remains NP-complete for planar
graphs [105], grid graphs with rectangular holes [98], and for (solid) grid graphs [97],
even if the node pairs are fully disjoint [98]. On grid graphs, the problem was shown to
be APX-hard recently [40].
Strongly related to both NDP and our approach is the subgraph homeomorphism
problem (SHP). There, the goal is to nd homeomorphic images of a pattern graph G
in a base graph H . A subgraph homeomorphism may be dened as a pair of mappings
(V, E). V maps nodes of G to nodes of H . For a node v of G, the mapping V(v) is called
its image. E maps edges of G to simple paths in H , for an edge e of G, E(e) is also called
its image. For an edge {u, v} in G, its image path E({u, v}) is required to start at V(u)
and end at V(v). Several variants of this problem exist. If the image paths are required
to be node-disjoint, and V is xed (then called the xed node-disjoint SHP), the problem
is equivalent to NDP. The NP-hardness of the general SHP has been observed early and
directly follows from a simple reduction from the Hamiltonian cycle problem [99]. The
main dierence between SHP and our approach is that our base graph is weighted, and
we are not looking for any subgraph homeomorphism, but for the optimal one (which
minimizes the total sum of edge weights). We also require the subgraph homeomorphism
to keep the original circular edge orderings at nodes.
Algorithmic aspects of orthoradial drawings have been investigated in [13] and [114].
5.3 Schematization on Octilinear Grid Graphs 187
In [13], Barth et al. showed that for a given planar input graph, a bend-free planar
orthoradial drawing has a combinatorial representation based on the angles of adjacent
nodes. Niedermann et al. [114] describe a polynomial-time algorithm to decide whether
such a representation is valid, and also describe a polynomial-time algorithm to obtain
such a representation. The complexity of nding such a representation with minimal
number of edge bends was left as an open question.
5.3 Schematization on Octilinear Grid Graphs
As mentioned above, our schematization approach is to nd an optimal image of G =
(V , E, L, L) in a specially crafted base grid graph. We denote this grid graph by Γ =
(Ψ, Ω). The grid cell size is D. Each v ∈ V is assigned an image node, which we denote
by V(v). Each e ∈ E is assigned an image path p = (ψ
0
,ψ
1
, . . . ,ψ
n−1
) through Γ, which
we denote by P(v). To preserve the topology of the input line graph, we require that
for any two input edges e and f , their paths P(e), P(f ) are node-disjoint (the rst and
last nodes might be shared if the original input nodes are equivalent). We additionally
require that the circular edge ordering of the input graph is preserved. In this section,
we describe our general approach by the example of an octilinear base grid and show
how an optimal such representation can be found both by an ILP (Section 5.4) and a fast
approximate approach (Section 5.5). Our goal is therefore to generate octilinear transit
maps. We will then extend our approach to other base grids in Section 5.10.
Denition 5.1 (Octilinear Grid Graph). We call a graph Γ = (Ψ, Ω) an octilinear grid
graph if it has been constructed as follows: for each position (x, y) on an N · M grid, a
grid node ψ
x,y
is added. For an ψ
x,y
, N
0
(ψ
x,y
) denotes the north neighbor ψ
x,y+1
, N
1
(ψ
x,y
)
denotes the north-east neighbor ψ
x+1,y+1
, and so on. Each ψ
x,y
is connected by an edge
to each of its neighbors N
0
(ψ
x,y
), . . . , N
7
(ψ
x,y
), if they exist.
A simple path p = (ψ
0
,ψ
1
, . . . ,ψ
n−1
),ψ
i
∈ Ψ in such an octilinear grid graph Γ then
describes an octilinear curve. Figure 5.6, left gives two examples. To optimize soft con-
straint S2, we strive for a minimum length of an image path p(e). We therefore only
consider shortest paths through Γ and assign each edge a hop cost w
h
. The total cost of a
path p = (ψ
0
,ψ
1
, . . . ,ψ
n−1
) through Γ is then
c(p) = (n − 1) ·w
h
. (5.1)
5.3.1 Line Bend Penalties
We again consider Figure 5.6, left. A shortest path in an octilinear grid graph might
contain a number of unnecessary bends. For example, the vertical and diagonal seg-
ments of path {t, u} in Figure 5.6, left can be distributed more optimally: path {t, u} in
188 Chapter 5. Schematic Transit Maps
𝑢
𝑣
𝑤
𝑡
Γ
𝑢
𝑡
𝑣
𝑤
Γ
0
Figure 5.6: Left: Two node-disjoint shortest paths {u, t } and {w, v} on an octilinear grid
graph Γ with uniform weights. Right: Same shortest paths on an undirected octilinear
grid graph Γ
0
extended by explicit bend edges, with higher costs for acute bends.
Figure 5.6, right has exactly the same number of vertical and diagonal edge segments,
but also minimizes the number of edge bends required, and uses an obtuse angle at the
bottom of the grid. To consider soft constraint S1, we also want path costs to reect the
number and acuteness of bends. Each bend should be punished by a penalty depending
on its angle, either w
135
, w
90
or w
45
, with w
135
≤ w
90
≤ w
45
to favor obtuse bends. We
do not punish a straight pass through grid nodes and set w
180
= 0. We dene
B
ψ
=
(
ψ
1
,ψ
2
)
∈ Ψ
2
| ∃ψ
0
: ψ
1
∈ adj
(
ψ
0
)
∧ψ
2
∈ adj
(
ψ
0
)
(5.2)
to hold all node pairs (ψ
1
,ψ
2
) adjacent to a common node ψ
0
and dene the function
w
b
: B
ψ
7→ {w
180
, w
135
, w
90
, w
45
} (5.3)
to return the bend penalty between edges {ψ
1
,ψ
0
} and {ψ
0
,ψ
2
}. We would like our path
cost function from above (Equation 5.1) to also consider bends and hence look like this:
c
b
(p) = c(p) +
n−2
i=1
w
b
(ψ
i−1
,ψ
i+1
) (5.4)
= (n − 1) ·w
h
+
n−2
i=1
w
b
(ψ
i−1
,ψ
i+1
), (5.5)
where n = |p| is again the path length and w
h
a uniform grid edge weight.
As there are 28 unique edge combinations
1
which produce a bend at a single grid
1
There are 8 · 7 = 56 combinations of incoming and outgoing edges (we ignore the case where a path
leaves a node through the incoming edge because the path is simple). As our graph is undirected, we
consider the bend for edges (e, f ) to be equivalent to the bend for edges (f , e).
5.3 Schematization on Octilinear Grid Graphs 189
𝜓
0
𝑥,𝑦
𝜓
1
𝑥,𝑦
𝜓
3
𝑥,𝑦
𝜓
4
𝑥,𝑦
𝜓
5
𝑥,𝑦
𝜓
7
𝑥,𝑦
𝜓
𝑥,𝑦
90
◦
90
◦
𝜔
2
𝑥,𝑦
135
◦
𝜓
2
𝑥,𝑦
𝜔
0,2
𝑥,𝑦
𝜔
2,7
𝑥,𝑦
𝜔
2,6
𝑥,𝑦
45
◦
45
◦
𝜔
1,2
𝑥,𝑦
𝜔
2,5
𝑥,𝑦
𝜔
2,4
𝑥,𝑦
𝜔
2,3
𝑥,𝑦
𝜓
2
𝑥,𝑦
𝜓
6
𝑥,𝑦
180
◦
135
◦
(b)
(a)
→
Γ
Γ
0
(c)
(d)
Figure 5.7: Transformation of an octilinear grid graph Γ (a) into its extended version
Γ
0
(b). Each original grid node ψ
x,y
is extended by 8 port nodes ψ
0
x,y
, . . . ,ψ
7
x,y
(c). These
are connected to ψ
x,y
by sink edges ω
0
x,y
, . . . , ω
7
x,y
. Bend edges ω
i,j
x,y
(each corresponding
to a specic bend angle) connect each port node to its sibling port nodes (d).
node, we cannot model the edge weights in our original grid graph to t such a cost
function. We therefore extend our octilinear grid graph by explicit bend edges.
Consider Figure 5.7. From now on, we denote the set of original grid nodes (simply
called grid nodes) by Ψ
д
3 ψ
x,y
. Correspondingly, the original grid edges (simply called
grid edges) are denoted by Ω
д
. Each such grid node ψ
x,y
is now extended by 8 so-called
port nodes ψ
0
x,y
, . . . ,ψ
7
x,y
, where port node ψ
i
x,y
leads to the corresponding port node of
neighbor N
i
(ψ
x,y
). We denote the set of port nodes by Ψ
p
. The original edges from ψ
x,y
to its neighbors are removed. To still be able to reach ψ
x,y
, the port nodes are connected
to their grid node by so-called sink edges ω
0
x,y
, . . . , ω
7
x,y
, so ω
i
x,y
= {ψ
i
x,y
,ψ
x,y
}. We denote
the set of sink edges by Ω
s
. Additionally, each port nodeψ
i
x,y
is connected to its clockwise
succeeding sibling ports i + 1, . . . , 7−i by so-called bend edges ω
i,j
x,y
. The set of bend edges
is denoted by Ω
b
. Each bend edge ω
i,j
x,y
then corresponds to a traversal of ψ
x,y
with a bend
of either 180
◦
, 135
◦
, 90
◦
or 45
◦
. We call the resulting graph the extended grid graph and
denote it by Γ
0
= (Ψ
0
, Ω
0
).
5.3.2 Modeling Edge Weights in the Extended Grid Graph
To arrive at the path cost function from Equation 5.5, the edge weights in Γ
0
have to
be modeled carefully. In particular, we must prevent shortcuts undermining our desired
cost model.
190 Chapter 5. Schematic Transit Maps
(1)
(2)
(3)
(4)
Figure 5.8: Shortcuts in traversing an extended grid node. (1) A 45
◦
bend replaced by
two sink edges. (2) A 90
◦
bend replaced by two sink edges. (3) A 45
◦
bend replaced by a
135
◦
bend followed by a 180
◦
bend. (4) A 90
◦
bend replaced by two 135
◦
bends.
𝜓
2,0
𝜓
1,0
𝜓
0,1
Γ
𝜓
0,1
𝜓
1,0
𝜓
2,0
Γ
0
Figure 5.9: Path (ψ
0,1
,ψ
1,0
,ψ
2,0
)
through the original grid graph
Γ and its extended grid graph Γ
0
.
Because sink edges are osetted
by a w
s
guaranteed to be greater
than any bend cost weight, sink
edges will only appear at the be-
ginning and at the end of a short-
est path from grid node to grid
node.
Sink Edge Weights
A path traversing through a grid node ψ
x,y
might choose to use two sink edges instead of
the corresponding bend edge (Figures 5.8.1 and 5.8.2). To avoid this, we dene a uniform
sink cost oset w
s
in such a way that it is always more expensive than any bend edge.
This is e.g. satised if we set w
s
= w
45
+ 1. Given a shortest path p
0
from a grid node ψ
x,y
to another grid node ψ
x
0
,y
0
in Γ
0
, it is then guaranteed that only the rst and the last edge
of p
0
are sink edges. The rst sink edge is used to leave ψ
x,y
, the last sink edge is used
to arrive at ψ
x
0
,y
0
(Figure 5.9). Any sink edge appearing in between would compromise
optimality, as it could be replaced by a cheaper bend edge.
Bend Edge Weights
Another issue can be seen in Figures 5.8.3 and 5.8.4: in a path traversing through a grid
node, both a 45
◦
and a 90
◦
bend edge may be replaced by two cheaper bend edges. A
45
◦
bend edge can be substituted by a 135
◦
bend edge followed by a 180
◦
bend edge
(Figure 5.8.3). As we want 180
◦
bends to have no cost, this would essentially mean that
any 45
◦
bend may be performed at the same cost as a 135
◦
bend. Analogously, a 90
◦
bend
edge may be replaced by two 135
◦
bend edges (Figure 5.8.4).
5.3 Schematization on Octilinear Grid Graphs 191
To block such shortcuts, we add a constant oset a ≥ 0 to the bend cost and denote
these updated bend costs by w
0
180
= w
180
+ a, w
0
135
= w
135
+ a, w
0
90
= w
90
+ a and
w
0
45
= w
45
+ a. To ensure the integrity of our original bend costs, we must choose a so
that the following inequalities hold:
w
0
135
+ w
0
180
= 2a + w
135
+ w
180
≥ a + w
45
= w
0
45
, (5.6)
2w
0
135
= 2a + 2w
135
≥ a + w
90
= w
0
90
. (5.7)
Inequality 5.6 ensures that substituting a 45
◦
bend edge with a 135
◦
bend edge followed
by a 180
◦
bend edge is never cheaper than the original 45
◦
bend. Inequality 5.7 ensures
that substituting a 90
◦
bend edge with two 135
◦
bend edges is never cheaper than the
original 90
◦
bend. As we assume that w
180
= 0 ≤ w
135
≤ w
90
≤ w
45
, the inequalities
hold for a = w
45
−w
135
:
2(w
45
−w
135
) + w
135
+ w
180
≥ w
45
−w
135
+ w
45
(5.8)
⇔ 2w
45
−w
135
+ w
180
≥ 2w
45
−w
135
(5.9)
⇔ −w
135
+ w
180
≥ −w
135
(5.10)
⇔ w
180
≥ 0, (5.11)
and
2(w
45
−w
135
) + 2w
135
≥ w
45
−w
135
+ w
90
(5.12)
⇔ 2w
45
≥ w
45
−w
135
+ w
90
(5.13)
⇔ w
45
≥ w
90
−w
135
. (5.14)
We note that all other possible bend edge replacements (for example, a 90
◦
bend might
be replaced by two 45
◦
bends) are already more expensive in our original cost structure,
and remain so regardless of the oset a. To also prevent sink edge shortcuts as described
above, we have to add a to the previously proposed w
s
.
We now again consider a shortest path p
0
on our extended grid graph Γ
0
from and
to an original grid node. For each ψ ∈ Ψ, we denote by ψ
∗
∈ Γ the original grid node ψ
belongs to (for port nodes, this is the corresponding grid node, for grid nodes, this is the
node itself). The following holds:
Lemma 5.1. A shortest path p
0
= (ψ
0
,ψ
1
, . . . ,ψ
n
0
−1
) through Γ
0
with ψ
∗
0
= ψ
0
and ψ
∗
n
0
−1
=
ψ
n
0
−1
(the rst and last node of p
0
are original grid nodes) always describes a simple path
p = (ψ
∗
0
,ψ
∗
2
,ψ
∗
4
, . . . ,ψ
∗
n
0
−1
) on Γ, with |p| = n = |p
0
|/2 = n
0
/2 (for each original grid node
visited by p, p
0
uses two grid nodes).
Proof. As any ψ ∈ Ψ
0
has a corresponding original grid node ψ
∗
, the existence of a
corresponding path p through Γ is easy to see. The start and end node of p will use
192 Chapter 5. Schematic Transit Maps
the corresponding original grid node and a port node to leave or reach the node. The
intermediate grid nodes of p will use a port node to reach the corresponding node in p
0
,
and a second port node to leave it. W.l.o.g., we ignore the case where a bend edge is
substituted by two or more bend edges of similar cost (as in the examples above), as this
neither aects the cost of p
0
, nor the course of the corresponding path p. A shortest path
p
0
through Γ
0
thus always visits exactly two port or sink nodes belonging to an original
grid node, or there is a corresponding shortest path of equivalent cost who does so. As p
0
is a shortest path, the corresponding path p = (ψ
∗
0
,ψ
∗
2
,ψ
∗
4
, . . . ,ψ
∗
n
0
−1
) is also simple.
We now set w
0
h
= w
h
−a. Then the following holds:
Lemma 5.2. Given a shortest path p
0
through an extended grid graph Γ
0
with edge weights
as described so far. The path p
0
goes from an original grid node ψ
x
1
,y
1
to an original grid
node ψ
x
2
,y
2
and has a corresponding path p in Γ. If the cost c
0
(p
0
) of p
0
is optimal (if p
0
is a
shortest path in Γ
0
), then p optimizes the desired cost function c
b
(p) from Equation 5.5.
Proof. According to Lemma 5.1, such a path p will always exist, and |p| = n = |p
0
|/2 =
n
0
/2. The cost of p
0
is
c
0
(p
0
) =
sink edges
2w
s
+
(
n − 1
)
·w
0
h
grid hops
+
osetted bend costs
n−2
i=1
a + w
b
ψ
∗
2i−2
,ψ
∗
2i+2
(5.15)
= 2w
s
+ (n − 1) · (w
h
−a) +
n−2
i=1
a + w
b
ψ
∗
2i−2
,ψ
∗
2i+2
(5.16)
= 2w
s
+ (n − 1) ·w
h
−a +
n−2
i=1
w
b
ψ
∗
2i−2
,ψ
∗
2i+2
(5.17)
= c
b
(p) + 2w
s
−a. (5.18)
Note that w
s
and a are positive constants. As a shortest path p
0
through Γ
0
thus minimizes
c
b
(p) + 2w
s
−a, it also minimizes c
b
(p).
Negatively osetting the uniform grid cost w
h
by a restricts the value range of w
h
.
Namely, it must hold that w
h
≥ a = w
45
−w
135
, as we would get negative edge weights
otherwise. This prevents for example the intuitive cost assignment of w
h
= 1, w
135
= 1,
w
90
= 2 and w
45
= 3. In our experiments, we will always use the cost assignment
w
h
>= 1, w
135
= 1, w
90
= 1.5 and w
45
= 2. Then, w
h
≥ w
45
−w
135
= 1.
5.3 Schematization on Octilinear Grid Graphs 193
5.3.3 Full Optimization Problem
So far, our path cost function c from Equation 5.5 only considers bends that occur along
paths between image grid nodes. This ignores bends that may occur between parts of the
same line passing through an input node v. To consider this, we also want to minimize
the sum c
v
b
of bend costs between adjacent image paths p(e)and p(f )where e and f share
at least one line (we say p(e) and p(f ) are adjacent if they share rst or last nodes). If
they share more than one line, we multiply the bend cost accordingly. Given for example
such paths p(e) = (ψ
0
, . . . ,ψ
n
) and p(f ) = (ψ
0
0
, . . . ,ψ
0
m
) with ψ
0
= ψ
0
0
, the bend cost
between p(f )and p(e)is w
b
(ψ
1
,ψ
0
1
). We can now dene the problem of nding an optimal
octilinear drawing of G on Γ
0
as follows:
Denition 5.2 (Optimal Metro-Map Image). Given a line graph G and an octilinear grid
graph Γ. For each input nodev ∈ V , nd an optimal assignment of image grid nodes I(v)
together with an optimal assignment of image grid paths P(e) for each input edge e ∈ E,
if such an assignment exists. We require that image grid paths are non-intersecting and
that circular edge orderings at v are preserved inψ (v)(hard constraint H2). The solution
should minimize (1) the sum of all path costs c
b
(P(e)), and (2) the sum of the distance
d
(
v,ψ (v)
)
between the original input node v and its image grid node ψ (v), weighted by
some w
m
(soft constraints S3), and (3) the sum c
v
b
as explained above.
More formally, the image (I, P) of G in Γ should optimize
t(I, P) =
e∈E
c(P
(
e
)
) +
v∈V
d
(
v, I(v)
)
w
m
+ c
v
b
. (5.19)
5.3.4 NP-Hardness
We continue to show that if the costs for edge bends and node displacements are part
of the input (as they are in our case), and if obstacles should be considered (which they
should in our case), our optimization problem is NP-hard. This proof, which is analogous
to the NP-hardness proof of the strongly related subgraph homeomorphism problem
given in [99], will serve as the motivation for the ILP described in the next section.
Lemma 5.3. The de cision variant of the Optimal Metro Map Image problem with obstacles
is NP-hard.
Proof. Given a grid graph with holes H with n nodes and m edges. We construct an
equivalent octilinear grid graph Γ by setting diagonal edge weights to innity and adding
obstacles corresponding to the holes of H . Horizontal and vertical edge weights are set
to 1. Additionally, we set edge bend costs to zero, thus eliminating the need to construct
the extended version of Γ. We now construct a graph G = (V , E, L, L) with n nodes,
connected in a cyclic fashion. As deg(v) = 2 for any v ∈ V , the circular edge ordering
194 Chapter 5. Schematic Transit Maps
Γ
G
H
Figure 5.10: The Hamilton path problem on a (general) grid graph H as a special case of
our Optimal Metro Map Image problem. A metro map image of G on Γ yields a Hamilton
path on H. Red grid edges were outtted with innite cost.
at input edges will always be preserved. There is a Hamiltonian cycle in H if and only if
there is a metro map image of G in Γ with cost smaller than innity. As the Hamiltonian
cycle problem on grid graphs with holes is NP-hard [88], so is our problem.
Lemma 5.4. The de cision variant of the Optimal Metro Map Image problem with obstacles
is also NP-complete.
Proof. Given a metro map image, we can obviously calculate its target function value
and check whether it satises the hard constraints in polynomial time. The Optimal
Metro Map Image problem is therefore in NP, and by Lemma 5.3 NP-complete.
An alternative proof to show the NP-hardness of Optimal Metro Map Image without
obstacles, but with special node displacement costs (such that grid node images are in
eect xed beforehand) was already mentioned in passing in [21]. It works by reduction
from the k-Node Disjoint Paths Problem. Consider the case with zero bend costs, innite
costs for diagonal grid edges, and unit costs for horizontal and vertical grid edges. For
a given grid graph Γ and some parameter k, we construct a graph G = (V , E, L, L) with
|E| = k and deg(v) = 1 for each v ∈ V (edges in E are pairwise node-disjoint). Therefore,
circular edge orderings at nodes are again always preserved. We choose displacement
costs in such a way that each input node v ∈ V is assigned exactly one grid node ψ
x,y
(e.g. by setting the displacement costs of v to innity for all other grid nodes). Deciding
whether there is an optimal metro map image with cost ≤ m, where m is the number of
horizontal and vertical edges, is then equivalent to the k-Node Disjoint Paths problem
on solid grid graphs with disjoint node pairs, known to be NP-complete [98].
5.4 Integer Linear Programming
Motivated by the NP-hardness result from above, we will tackle the Optimal Metro Map
Image problem in this section using integer linear programming. Our goal is to optimize
5.4 Integer Linear Programming 195
Equation 5.19, while adhering to hard constraints H1-H3. We now use a slightly modied
version of our extended grid graph: each (undirected) grid edge {ψ ,ψ
0
} is now modeled
as a pair of directed edges (ψ ,ψ
0
) and (ψ
0
,ψ ). For each such edge e (now directed) and
each input edge e, we then introduce a binary decision variable x
e,ω
. x
e,ω
should be 1 if
ω is a part of the image path P(e). Each x
e,ω
is then added to the objective function with
its cost as a coecient. The rest of this section describes how the hard constraints and
the search for shortest paths are modeled. For the latter, we build on the classic linear
program formulation of the shortest path problem.
5.4.1 Image Nodes and Station Displacement
To retrieve the image nodes for each input node v, we add a binary decision variable x
v,ψ
for each pair consisting of an input node v ∈ V and a grid node ψ ∈ Ψ
д
. If V(v) = ψ ,
then we want x
v,ψ
= 1, or 0 else. As V is injective, we have to ensure that (1) each input
node is assigned exactly one grid node, and (2) each grid node is assigned to at most one
input node. To guarantee (1), we add the following constraint:
∀v ∈ V :
ψ ∈Ψ
д
x
v,ψ
= 1. (5.20)
For (2), we consider the following scenarios: a grid node ψ may either be used as an
image for an input node, as a pass-through node on a path, or remain unused. We dene
Ω
b
ψ
to be the set of bend edges belonging to ψ . Then the following set of constraints
enforces this:
∀ψ ∈ Ψ
д
:
v∈V
x
v,ψ
+
e∈E
ω∈Ω
b
ψ
x
e,ω
≤ 1. (5.21)
Ifψ is used as an image for an input node, the rst sum is already one. Then, the decision
variables x
e,ω
for each adjacent bend edge and each input edge must be 0 to fulll the
constraint, making it impossible to reach ψ via a bend edge. Note that in this case, it
is still possible to leave or reach ψ via a sink edge. It will become clear below how we
ensure that sink edges may indeed only be used if ψ is settled as an image node.
We now consider a grid edge ω ∈ Ω
д
. Equation 5.21 also ensures that ω may only
appear in a single path p through Γ
0
. Any second path p
0
using ω would have to pass
through the adjacent extended grid nodes, but these are already blocked by p.
5.4.2 Edge Continuity
As each x
e,ω
appears in the objective function with its cost as a coecient, any assign-
ment of x
e,ω
to form the path of input edge e is guaranteed to be cost optimal. Let
e = {s, t}. To make sure that the x
e,ω
do indeed form a path from V(s) to V(t), we have
196 Chapter 5. Schematic Transit Maps
to enforce that the respective grid edges are connected and that they go from V(s) to
V(t). To guarantee continuity, it is enough to ensure that each path node except the
start and end node has the same number of outgoing and incoming edges. As we ensure
that original grid nodes are never part of a path (except the end and start node), we can
limit this requirement to port nodes and hence add the following constraints:
∀e ∈ E ∀ψ ∈ Ψ
p
:
ω∈adj
+
(ψ )
x
e,ω
−
ω∈adj
−
(ψ )
x
e,ω
= 0, (5.22)
To handle start and end grid nodes V(s) and V(t), we add the following constraints:
∀e ∈ E ∀ψ ∈ Ψ
д
: x
t,ψ
− 2x
s,ψ
+
ω∈adj
+
(ψ )
2x
e,ω
−
ω∈adj
−
(ψ )
x
e,ω
= 0. (5.23)
Outgoing edges are now always sink edges, as original grid nodes are not adjacent to
other edges in the extended grid graph. The main dierence to Equation 5.22 is then
that we are counting outgoing edges twice. An outgoing sink edge could now only be
regulated by two incoming sink edges. But then our path has split somewhere, which is
prevented by Equation 5.22. The constraint can thus only be fullled if ψ was assigned
to the start node s, and thus x
s,ψ
= 1. To counter any incoming sink edge, the only way
is to set x
t,ψ
= 1 and settle ψ as the image node for t. As hinted above, this additionally
ensures that sink edges may only be used by a path if the corresponding original grid
node is settled as the image of either the start or end input node.
Note that Equation 5.22 allows some additional constellations. For example, an out-
going edge may be countered by an incoming edge and by settling ψ as the image of t.
While such assignments would be feasible, they would not be optimal (why would we
need to leave the target grid node V(t) again?). However, if we allow grid edge weights
of 0, it may be possible for stray circular sub-paths ((ψ ,ψ
0
), (ψ
0
,ψ )) of cost 0 to appear.
We prevent this by the following additional constraints:
∀(ψ , ψ
0
) ∈ Ω
д
:
e∈E
x
e,(ψ ,ψ
0
)
+ x
e,(ψ
0
,ψ )
≤ 1. (5.24)
Alternatively, such stray circular paths may be removed in a post-processing step.
5.4.3 Topology Preservation
To meet hard constraint H2, we have to ensure that no two image paths intersect and that
the circular edge ordering around image nodes V(v) matches that of the corresponding
input node v. Equation 5.21 already ensures that paths never cross at grid nodes, as only
a single bend edge may be active per extended grid node, and only a single sink edge
may be used at the start and end image node of a path, making it impossible for a path
5.4 Integer Linear Programming 197
to arrive and leave via a sink edge.
Because of the diagonal grid edges, however, our octilinear grid graph is not planar.
Let Ω
d
be the set of diagonal grid edges and for a ω ∈ Ω
d
, ω
×
∈ Ω
d
is its intersect-
ing diagonal grid edge. To prevent path crossings at such edges, we add the following
constraints per diagonal grid edge:
∀ω ∈ Ω
d
:
e∈E
x
e,ω
+ x
e,ω
×
≤ 1. (5.25)
To preserve the original circular edge ordering, we would like to have variables δ
v,e
∈
{0, . . . , 7} holding the octilinear direction of input edge e at image node V(v), that is
the direction of the corresponding rst or last grid edge of P(e). Consider an input edge
e = {s, t }. At the grid image node V(s), Equations 5.23 and 5.21 guarantee that only a
single outgoing sink edge is used by the image path P(e). If we only consider the start
node s, we can therefore obtain the desired assignment with the following constraints:
∀e = {s, t} ∈ Ω :
ψ ∈Ψ
д
7
p=1
px
e,(ψ ,ψ
p
)
−δ
s,e
= 0. (5.26)
At grid node ψ , an outgoing sink edge (ψ , ψ
p
), p ∈ 0, . . . , 7 adds p to the sum, and as only
one such edge is used by P(e), δ
s,e
has to be set to the corresponding octilinear direction
to fulll the constraint. We add a similar constraint for incoming sink edges:
∀e = {s, t} ∈ Ω :
ψ ∈Ψ
д
7
p=1
px
e,(ψ
p
,ψ )
−δ
t,e
= 0. (5.27)
Let now u
0
, . . . , u
p
, . . . , u
deg(v)−1
be the clockwise ordering of nodes u
i
adjacent to
some v ∈ G. We reconsider the variable δ
v,e
dened above. Following an observa-
tion made in [115], we note that δ
v,(v,u
p
)
< δ
v,(v,u
p+1
)
must hold for all but one p ∈
{0, . . . , deg(v) − 1} if the original circular ordering was kept. We introduce a binary
helper variable β
p,v
and add the following set of constraints:
∀u ∈ V : δ
v,(v,u
p+1
)
−δ
v,(v,u
p
)
+ 8 · β
p,v
≥ 1. (5.28)
If δ
v,(v,u
p
)
< δ
v,(v,u
p+1
)
, the constraint is satised. The case were p “carries over” and
δ
v,(v,u
p
)
> δ
v,(v,u
p+1
)
is handled by setting β
p,v
to 1. To guarantee that this may only
happen once, we add the following simple constraint:
∀u ∈ V :
deg(u)−1
p=0
β
p,v
≥ 1. (5.29)
198 Chapter 5. Schematic Transit Maps
We only need to add the constraints from Equations 5.28 and 5.29 to input nodes u
with deg(u) > 2. The circular edge ordering of nodes adjacent to 1 or 2 edges is always
trivially preserved.
5.4.4 Line Bend Minimization
As explained in Section 5.3.2, line bends along paths are already minimized by our mod-
eling of the edge weights. To also penalize line bends between adjacent input edges in
the objective function, we would like to have binary decision variables ∆
p
e,f
for each
octilinear bend direction p between the image paths P(e) and P(f ). We already have
access to the outgoing and incoming angles of paths P(e) and P(f ) at their start and
end node via the variables δ
v,e
. For two such variables δ
u,e
and δ
u,f
, δ
u,e
−δ
u,f
mod 8 is
1 or 7 for 45
◦
bends, 2 or 6 for 90
◦
bends, 3 or 5 for 135
◦
bends and 4 for 180
◦
bends. We
denote this value by ∆
e,f
. Since modulo cannot be part of a linear program, we use the
following pair of constraints to get the desired value for ∆
e,f
. Each constraint is added
for each pair of adjacent input edges.
δ
u,e
−δ
u,f
+ 8γ
e f
≥ 0, (5.30)
δ
u,e
−δ
u,f
+ 8γ
e f
≤ 7. (5.31)
The auxiliary binary variable γ
e f
will be 1 if δ
u,f
> δ
u,e
per Equation 5.30, and only
then (Equation 5.31). We therefore get ∆
e,f
= δ
u,e
− δ
u,f
mod 8 = δ
u,e
− δ
u,f
+ 8γ
e f
.
As we want to assign dierent penalties to dierent bend angles, we introduce 8 binary
decision variables ∆
0
e,f
, . . . , ∆
7
e,f
, one for each possible value of ∆
e,f
. This assignment is
guaranteed by the following constraints, which are again added for each pair of adjacent
input edges.
∆
e,f
−
7
i=0
i∆
i
e,f
= 0. (5.32)
As this might result in multiple incorrectly activated decision variables (e.g. ∆
0
e,f
= 1 and
∆
5
e,f
= 1 if ∆
e,f
= 6), or no activated decision variable at all (if ∆
e,f
= 0), we additionally
ensure that exactly one ∆
i
e,f
is activated for each pair of adjacent input edges:
7
i=0
∆
i
e,f
= 1. (5.33)
Each ∆
i
e,f
is then added with the corresponding bend penalty to the objective function.
5.5 Approximate Solution via Iterative Shortest Path Calculation 199
5.4.5 Asymptotic ILP Size
We close this section with an analysis of the asymptotic size of our integer linear pro-
gram. The size is measured in terms of number of constraints and number of variables.
Given an input line graph G = (V , E) and a corresponding X ·Y grid. Our octilinear grid
graph contains Θ(XY ) edges, and Θ(XY ) nodes. As the extended version adds a constant
number of edges and nodes to each original grid nodes, the asymptotic number of edges
and nodes remains the same. There will be Θ(|V |XY ) station placement variables x
e,ω
(one for each pair of input node and original grid node) and Θ(|E|XY ) edge use vari-
ables x
e,ω
(one for each pair of input edge and extended grid edge). For the circular edge
ordering, we introduce two variables δ
v,e
for each input edge, so Θ(|E|) in total, and a
constant number of helper variables β
p,v
per input node, so Θ(|V |) in total. Likewise,
we require at most 8 · 7 helper variables γ
e f
per input node v for the line bends at input
nodes, as there are at most 8 ·7 pairings of edges per v. Finally, we need 8 bend variables
∆
i
e,f
per input node v and edge pair, so at most 8
2
·7 per v. The total asymptotic number
of variables in our ILP is therefore O(|E|XY + |V |XY ) = O(|E|XY ).
Regarding constraints, Equations 5.20 and 5.21 add Θ(|V |XY ) constraints. Equa-
tion 5.22 adds 8 · 7 constraints per original grid node and input edge, so Θ(|E|XY ) in
total. Likewise, Equation 5.23 adds 8 constraints per original grid node and input edge,
so again Θ(|E|XY ) in total. Equation 5.24 adds exactly one constraint per original grid
edge, so Θ(XY ) in total. Equation 5.25 adds Θ(XY ) constraints. Equations 5.26 and 5.27
add Θ(|E|) constraints. Equations 5.28 and 5.29 add Θ(|V |) constraints. Equations 5.30,
5.31, 5.32 and 5.33 all add at most 8 · 7 constraints per input node.
The total number of constraints in our ILP is therefore again O(|E|XY + |V |XY ) =
O(|E|XY ). As the grid size X ·Y depends on the bounding box area A of the input graph
G, both the number of constraints and variables in our ILP is O(|E| · dA/D
2
e).
5.5 Approximate Solution via Iterative Shortest Path
Calculation
Our experimental evaluation in Section 5.11 will show impractically high solution times
to solve the ILPs to optimality. In some cases, our ILP did not even nd a feasible solution
in under 12 hours. But even for simple input graphs, the solution times are too long
for our ILP to be used in an interactive setting. In this section, we introduce a fast
approximate approach that will show to work well in practice. Our basic method consists
of 4 steps:
1. Derive an ordering on the input edges E.
2. On the extended grid graph Γ
0
, iteratively calculate the shortest path P(e) from a
set S of start image node candidates to a set T of target image node candidates. We
200 Chapter 5. Schematic Transit Maps
1
2
4
3
7
1
3
5
1
2
4
4
1
1
6
5
6
8
9
10
11
Figure 5.11: Left: Input line graph G with an edge processing order (black) based on the
growth method: we select the node with highest line degree (red), process adjacent edges
in the order of the adjacent node’s line degree and mark the adjacent node as dangling.
Then we continue in this fashion from the dangling node of highest line degree. Right:
Metro map image after edge 7 has been routed.
call this process image path routing. Any path P(e
0
) which has already been settled
should not be crossed. The sink edge weights of S and T reect the node movement
penalty of the respective grid node candidate. We apply some smart heuristics to
avoid stalling of our approach.
3. Perform a local search to update node positions and re-route image paths.
5.5.1 Input Edge Ordering
To both nd a feasible and good solution, the order in which we process the input edges
is crucial, but it is not immediately evident which ordering is best. This section discusses
a few methods, which will be evaluated in Section 5.11.
Order by Number of Lines
To settle important network segments rst, our baseline idea is to order E in descending
manner by the number of lines |L(e)| traversing e. In a typical network, this will lead to
central segments being routed rst.
Order by Length
As network paths with a high station density are usually the most dicult to octilin-
earize, we may order E by the length of an edge e = {u, v} (induced by the geographical
distance between u and v) in ascending manner.
Order by Adjacent Node Degree
Another method of capturing the importance of a line graph edge e = {u,v} is the
(unordered) pair {deg(u), deg(v)} of the start and end node degrees. For an edge e =
5.5 Approximate Solution via Iterative Shortest Path Calculation 201
{u,v}, let o
deg
(e) = (max{deg(u), deg(v)}, min{deg(u), deg(v)}). We then say e >
deg
f if
o
deg
(e) > o
deg
(f ), where > is a lexicographic ordering relation.
Order by Adjacent Node Line Degree
We denote a node’s line degree ldeg(v) to be the number of (non-unique!) lines on each
adjacent edge (see Figure 5.11, left, for examples). For an edge e = {u, v}, let o
ldeg
(e) =
(max{ldeg(u), ldeg(v)}, min{ldeg(u), ldeg(v)}). We then say e >
ldeg
f if o
ldeg
(e) > o
ldeg
(f ),
where > again is a lexicographic ordering relation.
Growth-Based
Intuitively, it makes sense to let the map “grow” around a central node. We therefore
also consider the following ordering method: First, determine the node v with the high-
est degree (or line degree). In the order of one of the methods above, add the adjacent
edges to the ordered list of edges, add the adjacent nodes to a queue of “dangling” nodes,
and mark v as settled. From the list of dangling nodes, retrieve the unsettled node with
the highest degree (or, again, line degree) and continue in the same fashion. If no dan-
gling node is left, but unsettled nodes remain, the graph consisted of several connected
components, and we continue from the unsettled node with the highest (line) degree.
5.5.2 Edge Routing and Station Placement
After we have obtained an input edge ordering e
1
, . . . , e
i
, . . . , e
|E|
, we iteratively search
for a shortest image path P(e
i
) for each e
i
= {s, t}. To also determine the image grid
nodes V(s) and V(t) at the same time, we search for the optimal set-to-set shortest path
from a set S of source grid node candidates s, and a setT of target grid node candidates for
t. It can be computed by a variant of Dijkstra’s algorithm (see Section 1.3.3 for a more
detailed description). The image grid nodes are then settled to the start and end grid
nodes of the shortest path for each subsequent iteration. If a later image path is searched
for an edge e
j
and an adjacent node u has already been settled, then the corresponding
candidate grid node set U only contains V(u). To consider the displacement penalties
d
(
v, I(v)
)
w
m
for the image grid nodes, we add the respective displacement cost for s to
the sink edges at each ψ ∈ S, and the displacement costs for t to the sink edges at each
ψ ∈ T . Note that if S and T overlap, the shortest path (with zero cost) would always
be a single node in S ∩T . To prevent this, we ensure that S ∩T = ∅ by rst building
a Voronoi diagram: for each ψ
x,y
∈ S ∪T , move ψ
x,y
into S
0
if the geographic distance
of its position is closer to the position of s than the position of t. Otherwise, move ψ
x,y
into T
0
. Then use S
0
and T
0
instead of S and T . In case the image grid node for s or t has
already been settled in a previous iteration, we simply remove this grid node from the
other candidate set.
202 Chapter 5. Schematic Transit Maps
𝑡
𝑡
V (𝑠 )
?
?
𝑠
𝑇
𝑇
𝑆
𝑟
Figure 5.12: Left: To nd the image for an input edge {s, t } we must nd the shortest
path from a set of S of grid node candidates for node s to a set T of grid node candidates
for node t. We chose the candidates within a distance of r to the original input node
position. Right: If one of the nodes (here: V(s) ) was already settled, the image search
is a one-to-many shortest path calculation from V(s) to T .
During our experiments, we have found that using all original grid nodes as candi-
date image nodes for an input node v is excessive, as grid node candidates with high
displacement cost are almost never used in the nal metro map image. To speed up our
approach, we therefore used a radius r in our experiments to limit the size of our image
grid node candidate sets S and T (see Figure 5.12, left, for an example). This radius r
essentially species a maximal node displacement.
5.5.3 Topology Preservation
As in the ILP formulation, we have to ensure that paths do not cross and that circular
edge orderings are preserved (hard constraint H2). To achieve this, we update the edge
weights of the extended octilinear grid graph after an image path has been found. After
such a path P(e) = ψ
1
, . . . ,ψ
n
has been found, we set the weight of all bend and all sink
edges adjacent to any original grid nodeψ
∗
i
in the path to ∞. We say ψ
∗
i
is bend closed and
sink closed. This ensures that no original grid node may appear in any image path found
in a later iteration. To ensure that image paths are also edge-disjoint, we additionally
set the weight of each edge used in P(e) to ∞.
If a grid node ψ
x,y
= V(v) is sink-closed, its sink edges may be opened in a later
iteration which searches for an image path for some edge e = {v, u}. As explained
above, the sink edge weights will then be set to the displacement cost of ψ
x,y
for v. Also
note that in such a case, it will still remain bend-closed.
To ensure that paths do not cross at diagonal edges, we use the same technique as
5.5 Approximate Solution via Iterative Shortest Path Calculation 203
v
e
f
д
P(e)
P(f )
(1)
(2)
(3)
V(v)
ω
0
ω
1
ω
2
ω
3
V(v)
G
Γ
0
Figure 5.13: (1) An input line graph G. To respect the circular edge ordering around v,
we would like to block the area marked red in (2) for routing image path P(д). We achieve
this by setting sink edges corresponding to the octilinear directions of the blocked area
to innity (3).
in the ILP: Let Ω
d
be the set of diagonal grid edges and for a ω ∈ Ω
d
, let ω
×
∈ Ω
d
be its
intersecting diagonal grid edge. For each ω ∈ Ω
d
which is also part of P(e), we set the
weight of ω
×
to ∞.
It remains to guarantee that the original circular edge ordering is preserved. Con-
sider Figure 5.13. If for an input edge e = {v, u} an image path P(e) has already been
found in a previous iteration, and if we have now found an image path P(f ) for an edge
f = {v, w } which appears directly before or after e in the original circular edge order-
ing at v (Figure 5.13.1), we prevent any future image path from violating the original
circular edge ordering by closing the corresponding sector between P(e) and P(f ) (Fig-
ure 5.13.2). We do this by setting the weights of the corresponding grid edges to ∞ for
any future image path search for an edge adjacent to v (weights ω
0
...ω
3
in Figure 5.13.3).
5.5.4 Line Bend Minimization
To nally penalize line bends at input nodes, we oset the sink edges at an already
settled image grid node V(v) to reect the bend costs between the image path P(e) we
are currently searching for and any previously found image paths (Figure 5.14). Each
sink edge weight ω
i
is oset by the bend cost that would be incurred if sink edge ω
i
x,y
was used by P(e) (and would thus leave in octilinear direction i).
5.5.5 Stalling Prevention Heuristics
During image path routing, locally optimal routing decisions might make it impossible
to reach a feasible solution. We apply several locally informed heuristics to prevent such
stalling.
We now dene the circular distance o between two edges e and f adjacent at v as
204 Chapter 5. Schematic Transit Maps
v
e
f
д
(1)
(2)
P(д)
P(f )
(3)
c
45
c
135
+
∞
∞
∞
c
90
c
180
+
c
135
c
135
+
c
180
c
90
+
c
135
c
45
+
V(v)
V(v)
V(v)
G
Γ
0
Figure 5.14: (1) Input node v with adjacent edges e, f , д. (2) Grid edge with image paths
P(д) and P(f ) already routed. (3) Sink edge weight osets at V(v) prior to routing an
image path P(e) with e = {v, u}. Sink edge weights reect the bend penalty between
lines on e and lines on д and f . Edges blocked for P(e) to preserve the topological edge
ordering as explained in Section 5.5.3 are marked red.
𝑣
𝑒
𝑓
𝑔
𝑜 (𝑒, 𝑔, 𝑣) = 2
P(𝑒)
P(𝑔)
1
(1)
(2)
(3)
𝜔
0
𝜔
1
𝜔
2
V (𝑣)
𝐺
Γ
0
V (𝑣)
𝑜 (𝑒, 𝑓 , 𝑣) = 1
Figure 5.15: (1) A node v with 3 adjacent edges e, f , д for input line graph G. The blue
arrows give the circular edge ordering distance for edge e. (2) Image path P(e) has been
routed to leave V(v) via sink edge ω
0
, P(д) has been routed to leave V(v) via sink edge
ω
1
, leaving no space to route P(f ) without violating the circular edge ordering. (3) Sink
edges ω
0
and ω
1
are blocked before P(д) is routed to ensure enough free sink edges.
5.5 Approximate Solution via Iterative Shortest Path Calculation 205
u
(1)
(2)
Γ
0
w
v
V(v)
V(u)
V(w)
(3)
Γ
0
v
G
e
f
Figure 5.16: (1) Input line graph G. (2) The situation after image path P(e) has been
routed. There is now no path to image node V(v) which is node-disjoint with already
routed image paths. (3) To avoid situations like these, we drop grid nodes with free
degree lower than deg(v) as image candidate nodes for v.
the distance between e and f in the clockwise circular edge ordering of v, going in the
direction from e to f . For example, if the clockwise circular edge ordering at a node v is
e, f , д, h, then o(e, f , v) = 1, o(e, e, v) = 0, o(e, д, v) = 2, o(e, h, v) = 4, o (h, e,v) = 1, etc.
Consider Figure 5.15 and assume the input edge ordering at v has been determined to be
e, д, f . Edges e and д have already been routed. In the input graph, the circular distance
between д and e is o(д, e,v) = 2, and o(e, д, v) = 1 (Figure 5.15.1). In the extended
octilinear grid graph, the optimal image path P(д) for д leaves V(v) at a 45
◦
angle,
reducing the circular distance between P(д) and P(e) to 1 (Figure 5.15.2). This makes it
impossible to route f in a such a way that the original circular edge ordering is respected,
as it would have to leave between P(e) and P(д) - but there is no grid edge left. To
prevent such situations, we ensure that each image path will always leave enough free
grid edges in both directions by setting the weight of sink edges corresponding to the
octilinear direction which would violate this constraint to innity (Figure 5.15.2).
Another cause for stalling can be seen in Figure 5.16. Assume the green edges con-
necting u and w have already been routed, as has the orange edge e connecting v. Rout-
ing P(e) has determined V(v) to be the optimal grid node for v (Figure 5.16.2). This,
however, makes it impossible to route the blue edge f , as V(v) cannot be reached any-
more. To avoid such stalling situations, we exclude grid edges with a free node degree
smaller than the node degree of input node v from the grid candidates (Figure 5.16.3).
The free node degree is the number of adjacent grid nodes that have not yet been blocked.
5.5.6 Complexity
The complexity of our approximate approach basically traces back to |E| shortest path
searches through Γ
0
. To nd an initial ordering of the input edges, we have to sort
|E| input edges, which can be done in O(|E| log |E|). To assign each v the set of node
candidates with their distance penalties, O(|V | · |Ψ|) distance calculations are required.
206 Chapter 5. Schematic Transit Maps
V
0
(v )
(1)
(2)
(3)
V
0
0
(v )
V
0
1
(v )
V
1
(v )
V
2
(v )
Figure 5.17: Three steps during our local search.
To nd the images paths, we require |E| shortest paths searches on the extended grid
graph Γ
0
. The number of nodes |V
0
| in Γ
0
is Θ(|V |), and the number of edges |E
0
| in Γ
0
is
Θ(|E|), whereV and E are the nodes and edges in the original octilinear grid graph. Using
Dijkstra’s algorithm, the shortest path searches therefore take O(|E| ·(|Ω| + |Ψ| log |Ψ|))
time. After each Dijkstra run, all grid nodes that appear in the path have to be both bend
closed and sink closed. In the worst case, this takes time O(|Ψ|),
Additionally, we have to update the sink edge weights at the image grid nodes after
each Dijkstra run. To update the line bend costs, we have to check whether two adjacent
edges share a common line. We assume that the lines are given in such a way that we can
intersect two sets L (e) and L(f ) in time linear in the bigger list (this would be possible if
lines are identied by an integer and line sets are sorted.) Then let M again be the total
number of unique lines in the line graph. As the maximum input node degree is 8 and
the number of line pairs we have to check is upper bounded, the line bend updates per
image node take time O(M),
In total, we need O(|E| · (|Ω| + |Ψ| log |Ψ + M)) time. Just as for the ILP size, |Ψ| ∈
O(|Ω|) = O(XY ). Therefore, the overall complexity of our approach is O(|E| · (|Ω | +
|Ψ| log |Ψ| + M)) = O(|E| · (XY + XY log XY + M)) = O(|E|XY · (1 + log XY ) + M) =
O(|E|·A/D
2
·log A/D
2
+ M) (just as for the asymptotic ILP size, A is the area of the input
line graph and D is the grid cell size). Just as for the complexity of the edge untangling
rules in Chapter 4, M may be considered a constant factor.
5.5.7 Local Search Optimization
Our approximate approach allows for an additional straightforward local search opti-
mization step. Figure 5.17 gives an example. Given an initial feasible metro map image
(V
0
, P
0
) on a grid graph Γ. We say the set of all metro map images (V
0
0
, P
0
0
) where
exactly one input node v is moved from V
0
(v) to one of the 8 adjacent grid nodes
N
0
(V
0
(v)), . . . N
7
(V
0
(v)) (if free) and the image paths for all adjacent edges are rerouted
is the search neighborhood for (V
0
, P
0
). For each neighboring metro map image, we cal-
culate the objective function and take the best (V
0
0
, P
0
0
) as (V
1
, P
1
).
5.6 Speedup Techniques 207
|P(𝑒 )| = 2
|P(𝑒 )| = 3
V (𝑣 )
V (𝑢)
𝐹
𝑢,𝑣
|P(𝑒 )| = 4
𝐹
𝑢,𝑣
V (𝑢)
V (𝑢)
V (𝑣 )
V (𝑣 )
P(𝑒 )
P(𝑒 )
P(𝑒 )
Figure 5.18: Spring-embedding used to enforce a uniform distance between re-inserted
nodes during our local-search phase if the degree-2 heuristic was applied. Left: Image
path P(e) is not long enough to hold the k = 2 contracted nodes without violating
the minimum-distance constraint. The spring stores the potential energy of E
u,v
=
1
4
c,
where c is a penalty parameter. Middle: the edge P(e) is unnecessarily long, again with
a potential energy of E
u,v
=
1
4
c. Right: V(u) and V(v) have been chosen in such a way
that the spring is relaxed.
5.5.8 Distance Optimization Between Contracted Degree-2 Nodes
So far, we have ignored that the application of the degree-2 heuristic may lead to very
small distances between the re-inserted nodes in the nal octilinear drawing. In partic-
ular, hard constraint S2 may be violated. While this is sometimes employed in manually
created transit maps to save space, extremely small distances between them may lead to
overlapping station markers and/or labels. On the other hand, if the distances between
them become too great, the resulting lines in the nal drawing may appear unnaturally
elongated. To address these issues and still keep some exibility regarding the distance
between adjacent degree-2 nodes, we use a spring-based approach. Given an input edge
e = {u, v} and its image path P(e). Let |P(e)| be the number of original grid edges in the
path, and let k be the number of nodes on e that have been contracted before schemati-
zation. We dene a spring force F
u,v
=
c
k
· (k + 1 − |P(e)|) between nodes u and v. The
penalty factor c is a parameter of our approach. If |P(e)| = k + 1, we can insert all k
contracted nodes in such a way that their positions correspond to an original grid node,
and the spring is relaxed. To consider the spring in our local search phase, we add the
potential energy E
u,v
=
c
2k
· (k + 1 −l)
2
to our objective function.
5.6 Speedup Techniques
Apart from general implementation tuning, there are several algorithmic techniques that
can be employed to further speed up our approach. This section describes a general A
∗
heuristic to speed up the image path routing as well as several methods to prune the
octilinear base grid. While the former is obviously only applicable to our approximate
approach, the latter can also be used to reduce the size and solution times of our ILP.
208 Chapter 5. Schematic Transit Maps
ψ
0
ψ
1
3
3
1
1
ψ
0
ψ
1
3
1
1
(1)
(2)
ψ
1
ψ
0
3
3
1
1
ψ
0
ψ
1
3
1
1
(3)
(4)
(5)
ψ
0
ψ
1
1
1
1
1
1
1
Figure 5.19: (1, 2, 3, 4): If diagonal costs are lower than horizontal or vertical cost, one
or two horizontal or vertical edges (blue) may be replaced by cheaper combinations (red)
involving a diagonal edge. (5) Example where the Chebyshev distance is a bad heuristic:
the shortest path distance between ψ
0
and ψ
1
is 2 (using the diagonal red edges), but the
Chebyshev estimates 4 by only considering vertical and horizontal movements (blue).
5.6.1 Shortest-Path Heuristic
In a classic grid graph with diagonal edges and unit edge weights, the L
∞
metric or
Chebyshev distance D
Ch
((x, y), (x
0
, y
0
)) = max{|x −x
0
|, |y −y
0
|} between two grid nodes
ψ
x,y
and ψ
x
0
,y
0
is an admissible A
∗
heuristic for calculating shortest paths. This is evi-
dently true, as we always need at least D
Ch
((x, y), (x
0
, y
0
) edges for a simple path from
ψ
x,y
toψ
x
0
,y
0
. As obstacles may only elongate the shortest path fromψ
x,y
toψ
x
0
,y
0
, this still
holds on octilinear grid graphs with holes or when already routed paths act as obstacles.
From Equation 5.5 and Lemma 5.2, it is clear that c(p) ≤ c
b
(p) ≤ c
0
(p). Any admissi-
ble heuristic for the cost function c(p) is therefore also an admissible heuristic between
original grid nodes on the extended grid graph. If we consider unit weights for origi-
nal grid edges, the Chebyshev distance therefore remains an admissible heuristic there,
even if bend and sink costs are considered (intuitively, bend costs may only elongate the
shortest path, and sink costs only increase the shortest path costs.) However, we would
like our approach to allow for exible penalties for dierent bends angles, as well as for
using diagonal, horizontal and vertical grid edges. It is clear from above that dierent
bend penalties won’t change the admissibility of a Chebyshev distance heuristic.
Let w
v
h
be a uniform vertical grid edge weight, w
h
h
a uniform horizontal grid edge
weight and w
d
h
a uniform diagonal grid edge weight. We want our heuristic to reect
these grid edge weights. However, as diagonal, horizontal and vertical edges might be
replaced by cheaper combinations of other grid edge types, it is not immediately clear
how these weights should be considered. In particular, we want to avoid the trivial
admissible method to just multiply the Chebyshev distance by min{w
d
h
, w
v
h
, w
h
h
}, as this
might result in a bad heuristic. Instead, we use the following approach: For a path from
ψ
x,y
to ψ
x
0
,y
0
, we consider two cases: (1) x = x
0
or y = y
0
, (2) x , x
0
and y , y
0
. As a
basic “currency”, we then use admissible weights w
h
h
and w
v
h
. The admissible horizontal
5.6 Speedup Techniques 209
weight w
h
h
is chosen in such a way that for a shortest path from ψ
x,y
toψ
x
0
,y
(note that the
y coordinates are the same), |x −x
0
| ·w
h
h
≥ w
h
h
. Similarly, the admissible vertical weight
w
v
h
is chosen in such a way that for a shortest path from ψ
x,y
to ψ
x,y
0
, |y −y
0
| ·w
v
h
≥ w
v
h
.
To nd these admissible weights, we have to take shortcuts into account.
For w
h
h
, we have to consider two cases: either w
d
h
≥ w
h
h
, or w
d
h
< w
h
h
. In the former
case, we can never replace horizontal grid edges by cheaper variants. In the latter case,
however, two shortcuts appear: we may either replace two horizontal edges by two
cheaper diagonal edges (Figure 5.19.1), or we might replace a single horizontal edge by
a diagonal edge followed by a vertical edge (Figure 5.19.2). Note that in both cases,
w
h
h
= w
d
h
is admissible, regardless of the value of w
v
h
. Likewise, w
v
h
= w
d
h
is an admissible
weight for vertical edges (Figure 5.19.3 and 4). We therefore choose
w
h
h
=
w
d
h
if w
d
h
< w
h
h
,
w
h
h
else,
(5.34)
and
w
v
h
=
w
d
h
if w
d
h
< w
v
h
,
w
v
h
else.
(5.35)
If for two grid nodes ψ
x,y
to ψ
x
0
,y
0
we have both x , x
0
and y , y
0
, we also have
to account for diagonal edges that may be used to replace vertical and horizontal edges
(Figure 5.19.5). We additionally dene w
−d
h
to be the weight we might save by replacing a
vertical and a horizontal edge by a diagonal edge in such a case. This is straightforward:
w
−d
h
=
w
d
h
− (w
h
h
+ w
v
h
) if w
d
h
≤ w
h
h
+ w
v
h
,
0 else.
(5.36)
We can only replace a vertical and horizontal edge by a diagonal edge if the diagonal
weight w
d
h
is smaller than w
h
h
+ w
v
h
. If we do that, we save w
h
h
+ w
v
h
at the cost w
d
h
. An
admissible A
∗
heuristic for our extended octilinear grid graph with exible bend penalties
and exible grid edge weights w
v
h
, w
h
h
and w
d
h
is therefore
h(ψ
x,y
,ψ
x
0
,y
0
) = |x −x
0
| · w
h
h
+ |y −y
0
| · w
v
h
− min{|x −x
0
|, |y −y
0
|} ·w
−d
h
. (5.37)
In practice, we are searching for the shortest path into a set of target grid node can-
didates T . The heuristic then becomes h
0
(ψ
x,y
,T ) = min
ψ
x
0
, y
0
∈T
h(ψ
x,y
,ψ
x
0
,y
0
). Although
h(ψ
x,y
,ψ
x
0
,y
0
)is easy to compute, we would like to avoid calculating the heuristic for each
target node on each Dijkstra relaxation. To reach any node ψ
x
0
,y
0
∈ T , we must rst reach
a grid node which is part of the hull of T . The grid node with the minimum heuristic
value will be found on this hull. This hull can be quickly precomputed by selecting only
grid nodes ψ
x
0
,y
0
∈ T which are also adjacent to a node ψ < T .
210 Chapter 5. Schematic Transit Maps
ˆ
d
G
(1)
H
(2)
(3)
Figure 5.20: (1) Input line graph G on a full octilinear grid graph covering its padded
bounding box, and the convex hull H of the input nodes. (2) optimal metro-map image
of G on the full octilinear grid. (3) optimal metro-map image of G on an octilinear grid
cropped to H.
5.6.2 Sparse Base Grids
As both the asymptotic size of our ILP and the complexity of our approximate approach
mainly depend on the base grid size, we would like to study the eects of simplied
versions of the octilinear base grid. Figure 5.20 shows an input line graph G, a corre-
sponding extended octilinear grid graph, and the metro map image of G. It is obvious
that large parts of the grid graph may be pruned without aecting the metro map image.
In this section, we present three methods to reduce the size of the octilinear grid
graph: the rst one is straightforward and does not use the bounding box of G to con-
struct the grid graph, but the convex hull. The second approach is based on quadtrees
and aims to automatically adjust the grid graph density to the line graph density. The last
and most powerful approach extends the concept of a Hanan grid (originally conceived
in the context of Steiner trees) to an octilinear Hanan grid.
Padded Convex Hull of Input Graph
Let S be the set of coordinates of the input line nodes V . We pad each p = (x, y) ∈ S by
adding 4 additional coordinates p
1
. . . p
3
to S: p
1
= (x +d, y), p
2
= (x −d, y), p
3
= (x, y +d)
and p
4
= (x, y −d), where d is a padding parameter. We take the convex hull of P and
only keep the grid nodes that are within the hull. These grid nodes already induce an
octilinear grid graph with holes, which we use for octilinearization, as shown in Fig-
ure 5.20.3. As our method explicitly allows for obstacles, our approach so far (including
the A
∗
heuristic) still works unchanged.
5.6 Speedup Techniques 211
𝐺
(1)
𝐺
(2)
(3)
Figure 5.21: (1): An input line graph G and its quadratic padded bounding box. (2):
Quadtree constructed from the nodes of G. (3): Octilinear grid graph constructed from
the quadtree, and a metro-map image of G in it.
Quadtrees
A quadtree recursively subdivides a rectangle into four smaller rectangles of equal size
(quadrants), called cells, until a stopping criterion is met. Given an input line graph G, we
build an initial rectangle from the bounding box of G and extend it such that the height
and widths are even multiples of our grid cell size D (Figure 5.21.1). We then recursively
divide the cells until each cell contains exactly one input node (Figure 5.21.2), or until the
cell size D is reached. The intersection points of the cell boundaries then act as nodes,
and the boundary nodes of each cell are connected by diagonal edges (Figure 5.21.3).
This octilinear graph is then extended by bend edges in the same way as our original
octilinear grid graph. Grid edges in the resulting octilinear graph may now have lengths
which are multiples of the grid edge lengths in the corresponding octilinear grid graph
with grid cell size D. To not distort path cost, we multiply the edge weights accordingly.
Octilinear Hanan Grids
Given a set of two-dimensional points S, a Hanan grid H(S) is a graph constructed in
the following way: (1) Draw vertical and horizontal lines through each p ∈ S. (2) Where
these lines intersect, add nodes. We extend the concept of a Hanan grid by also drawing
diagonal lines through each p ∈ S and call such a grid an octilinear Hanan grid.
Denition 5.3 (Octilinear Hanan Grid). Given a point set S on the plane, an octilinear
Hanan grid H
O
(S) = (W
O
, F
O
) is the graph constructed from S like follows: (1) Draw
vertical, horizontal and diagonal lines through each p ∈ S. (2) Add nodes w ∈ W
O
at
intersection points.
Let now S be again the set of input node coordinates. To maintain a minimum seg-
ment length in the nal map, we have to make sure that edges in H
O
(S) have a minimum
212 Chapter 5. Schematic Transit Maps
ˆ
d
G
(1)
S
0
ˆ
d
(2)
(3)
Figure 5.22: (1): An input line graph G on a regular grid covering its padded bounding
box. The input nodes are snapped to the nearest grid node and form the input set S
0
(blue). (2): Octilinear Hanan grid constructed from S
0
. (3): Metro-map image of G on the
octilinear Hanan grid constructed from S
0
.
length of D. We therefore do not construct the octilinear Hanan grid directly on S. In-
stead, we rst construct a regular grid with cell size D from the (padded) bounding box
B of the input graph. Afterwards, we snap each point in S to the nearest grid node (we
allow multiple p ∈ S to share a grid node). We build the octilinear Hanan grid from the
resulting point set S
0
. Additionally, we only keep nodes that fall within B.
For a full octilinear grid graph Γ = (Ψ, Ω) with grid cell size D built from B, W
O
⊆ Ψ .
However, just like with quadtrees, grid edge lengths may now be multiples of the corre-
sponding grid edge lengths in Γ. We therefore again multiply the edge weights accord-
ingly to preserve path costs. Any path P from s to t in H
O
(S) then has a corresponding
path in Γ from s to t yielding the same octilinear curve (P may, however, not be the
shortest path from s to t in Γ.)
5.6.3 Iterative Octilinear Hanan Grids
An interesting feature of octilinear Hanan grids is that we may increasingly approximate
the corresponding full octilinear grid Γ until H
O
(S) fully converges to Γ. To achieve this,
we rst construct an initial octilinear Hanan grid H
O
0
(S). We then take the nodes W
O
0
as a point set S
1
and construct an octilinear Hanan grid H
O
1
(S
1
). This process continues
until a maximum number of iterations is reached, or until the grids converge. Figure 5.23
gives an example.
5.7 Obstacles 213
Figure 5.23: Far left: Box-restricted octilinear Hanan grid constructed of 4 input points
(black). From second left to right: Octilinear Hanan grids constructed by taking the
points of the previous grid as input. After 3 iterations, we arrive at the full grid.
5.7 Obstacles
A common requirement in designing schematic maps is that they should match some
existing base map. This could be satellite imagery or an existing schematic representa-
tion of rivers, lakes, border, coastal lines, forest, or mountain ranges. See for example the
excerpt of the ocial long-distance network map of the Swiss Federal Railways shown
in Figure 5.24. As we may add arbitrary oset costs to grid edges, our approach di-
rectly allows for a consideration of such obstacles by setting the weight of grid edges
intersecting an obstacle to innity. Figure 5.25 gives an example.
5.8 Approximating Ge ographical Courses
Another common requirement in schematic maps is that segments should follow their
original geographical course to a certain extent, for example if they are used as an over-
lay for satellite imagery or if large distances between stations would result in an extreme
distortion of the course. This becomes particularly important if we apply the degree-2
heuristic prior to the octilinearization. The introduction to this work presented a real-
world example of an octilinear map where line segments follow their geographic course
between stations (Figure 1.4). If degree-2 nodes were contracted in this example before
schematization, most of the lines in this map would appear as mostly straight line seg-
ments between intersection or terminus nodes. To approximate geographical courses,
a straightforward adaption of our approach is sucient: instead of using constant grid
edge weights, we now use a weight function w(ω, e) = w
h
+ b · d(ω, e), where w
h
is
the original constant weight for a grid edge, b is a penalty parameter, and d(ω, e) is the
geographical distance between input edge e and grid edge ω.
214 Chapter 5. Schematic Transit Maps
Figure 5.24: Ocial long-distance network map of the Swiss Federal Railways. The
base map depicting schematic representations of the borders and lakes is used in several
such maps. The octilinear metro map has to respect these obstacles.
G
Figure 5.25: Left: A metro-map image for an input line graph G should be found which
respects the obstacles formed by a river, a lake (both blue), and a forest (green). Right:
Grid edges intersecting the obstacles (red) were outtted with innite weight. The re-
sulting metro-map image avoids the obstacles.
5.9 Hard Constraint Relaxation 215
Figure 5.26: Left: Input line graph G. Middle: A metro-map image violating topology
constraint H2. Right: The nal drawing after additional non-station nodes have been
inserted at the positions where the violation rst occurred. The octilinearized line graph
now allows for a crossing-free line ordering. In the nal rendering, the topology is not
impaired.
5.9 Hard Constraint Relaxation
Our approximate approach might fail to nd an initial feasible metro map image without
constraint violations. By relaxing hard constraint H2 we enable our local search phase to
better explore the search space. Recall that hard constraint H2 was enforced by closing
grid nodes used in a path P(e)by setting the weight of sink and/or bend edges to innity.
To relax hard constraint H2, we instead introduce a weight w
∞
which is set high enough
so that any path through Γ using only non-closed edges is cheaper than w
∞
. This is for
example satised if we set w
∞
= |Ω | ·(max{w
h
h
, w
v
h
, w
d
h
}+ max{w
180
, w
135
, w
90
, w
45
}). To
close an edge in the extended octilinear grid graph, we then use the weight w
∞
instead
of ∞. Then, if there are enough grid nodes, we are guaranteed to nd an initial metro
map image, albeit one that may have topology errors. However, our local search may
now nd a feasible image respecting hard constraint H2 by trying dierent grid position.
Interestingly, a violation of hard constraint H2 might not always result in a metro
map drawing with incorrect topology. In Figure 5.26, the image paths for the orange
and red line, the green and brown line, and the brown and blue line are all not node
disjoint and therefore violate hard constraint H2. However, the schematized line graph
(with 3 new non-station nodes at exactly those position where the topology constraints
where violated) allows for a crossing free line ordering, retaining the topology of the
original input graph. The readability of the nal map is not impaired by this (implicit)
line bundling.
216 Chapter 5. Schematic Transit Maps
G
Figure 5.27: Left: An input line graph G and a triangular grid corresponding to the
padded bounding box of G. Right: Metro-map image (blue) of G on the triangular grid
graph.
5.10 Non-Octilinear Layouts
Except for the A
∗
heuristic in Section 5.6.1 and the base grid simplication approaches
in Section 5.6.2, our approach so far did not require the base grid to be octilinear, or even
ortholinear. Both our ILP approach and our approximate approach directly translate to
other regular base grids. In this section, we describe two base grids to render hexalinear
and orthoradial metro maps.
5.10.1 Hexalinear Maps
In hexalinear metro map drawings, each segment has an orientation which is a multiple
of 60
◦
. If we take a triangular grid (a partitioning of the plane into equilateral triangles),
any path on the resulting grid will be hexalinear. By using a triangular grid as a base
grid we can thus produce hexalinear maps. Adapting the extended nodes described in
Section 5.3.2 to the hexalinear setting is straightforward. Just as in the octilinear setting,
shortcuts may appear in the expanded grid nodes (which now have 6 ports). We avoid
them using the same techniques as described in Section 5.3.2.
5.10.2 Orthoradial Maps
As mentioned in the introduction to this chapter, orthoradial metro maps have found
some media and research interest in recent years [13, 110, 114, 113]. A straightforward
application of our approach to this setting would be to use an orthoradial grid as a base
grid, as depicted in Figure 5.28, left. We call the concentric circles of this grid its rings.
Such a grid can indeed be interpreted as a rolled-up ortholinear grid as noticed in [13].
5.10 Non-Octilinear Layouts 217
G
Figure 5.28: Left: An orthoradial grid graph with 3 rings around a center node. With
increasing ring radius, the grid density decreases, leaving large empty areas. Middle: A
pseudo-orthoradial grid graph in which the the number of nodes on a ring double each
time the ring radius doubles, creating a more uniform grid density. Right: A metro-map
image (blue) on such a pseudo-orthoradial grid.
From an esthetic point of view though, such a grid is not optimal, as the grid density
decreases with the ring radius. This leaves large empty areas in the nal map.
Pseudo-Orthoradial Grids
We propose to use a slightly dierent kind of base grid which we call pseudo-orthoradial:
starting with an inner ring r
1
of radius d and b
1
= 8 equidistant nodes, we keep adding
rings r
i
with a distance d to the last ring r
i−1
and nodes at the intersection points of an
imaginary line drawn through the center and the corresponding node on r
i−1
(as in a
standard orthoradial grid). However, at rings r
2
, r
4
, ··· , r
2
j
, we double the number of
nodes, that is b
2
= 16, b
3
= 16, b
4
= 32, b
i
= 8 · 2
blog
2
(i)c
. Figure 5.28, middle, gives an
example.
The length of segments connecting two rings will obviously always by d. The circular
segments are guaranteed to have a length of at least
π
4
d. To strictly guarantee a minimum
segment length of d on the entire grid, the radius of the inner ring must be set to
4
π
d,
although we found in our experiments that this hardly matters in practice.
Given an input line graph G, we then chose as a center node the node with the highest
line degree and use a pseudo-orthoradial grid graph with a radius large enough to cover
the entire bounding box of G as a base grid graph. As maps based on concentric circles
often feature an additional center node [114], we also add such a node and connect it to
the 8 nodes on ring r
1
(see Figure 5.28).
Edge Weights in Pseudo-Orthoradial Grid Graphs
We base the grid edge weights in this pseudo-orthoradial grid graph on a uniform weight
for edges connecting two rings, and a base weight w
r
for ring segment edges. On the
innermost ring r
1
, all ring segments are weighted by w
r
. For all other rings r
i
, we set
218 Chapter 5. Schematic Transit Maps
Table 5.2: Line graphs used in our experimental evaluation. Under
¯
d we give the average
distance of adjacent input nodes. A is the area of the input graph in k m
2
. The maximum
input node degree is given as deg
max
. The size of the corresponding grid graph with cell
size D =
¯
d is given in number of nodes |Ψ| and number of edges |Ω|. The last 4 columns
give the number of nodes |V | and the number of edges |E| with and without the degree-2
heuristic.
grid graph raw deg-2 heur
¯
d A deg
max
|Ψ| |Ω| |V | |E| |V | |E|
Freiburg (FR) 0.4k 75.8 4 3.7k 15.8k 76 79 19 22
Vienna (VN) 0.8k 0.4k 5 5.5k 23.7k 98 104 19 25
Chicago (CG) 1.0k 1.1k 4 10.2k 43.9k 145 151 28 34
Sydney (SD) 1.5k 3.8k 4 14.5k 63.0k 178 183 31 36
Berlin (BE) 0.7k 0.6k 6 10.5k 45.6k 172 184 36 48
Sydney (ST) 0.5k 0.4k 4 12.2k 52.7k 197 205 43 51
London (LO) 1.1k 2.2k 7 18.0k 78.4k 276 313 89 126
New York (NY) 0.6k 1.9k 8 50.4k 221.2k 465 520 113 168
the ring segment weights to w
r
· a, where a is the ratio between a segment on r
i
and
a segment on r
1
, that is a =
8i
b
i
. As we assume that higher acuteness of a bend angle
means higher bend cost, the extended grid nodes will oer no shortcut edges that may
circumvent our bend penalty system.
5.11 Experimental Evaluation
We evaluated our approach in all its variants on 8 real-world input line graphs of in-
creasing sizes and complexity: the tram network of Freiburg (F), the subway network
of Vienna (V), the light rail network of Chicago (C), the light rail network of Stuttgart
(ST), the subway network of Berlin (B), the light rail network of Sydney (SD), the un-
derground network of London (L), and the subway network of New York (NY). For an
overview of their dimensions, see Table 5.2.
All but the London subway dataset (which was constructed from data provided in a
public GitHub repository
2
) were generated from raw schedule data using the pipeline
described so far. Where necessary, we manually xed small errors in the input data
to ensure that the experiments only measured the performance of our schematization
technique. For all datasets, we measured the performance of our ILP approach (Sec-
2
https://github.com/oobrien/vis/tree/master/tube/data
5.11 Experimental Evaluation 219
tion 5.11.1), the eect of dierent input edge orderings on our approximate approach
(Section 5.11.2), and the quality and speed of the latter (Section 5.11.3). In Section 5.11.4
we also evaluate the eect of the various base grid minimization methods discussed in
Section 5.6.2 on quality and speed. In Section 5.11.5, we experiment with base grids for
alternative layouts, as described in Section 5.10.
To demonstrate that the resulting schematic maps are visually pleasing, we present
various rendered maps for all input graphs. To show the exibility of our approach, we
also experiment with grid edge weights to avoid obstacles, to approximate geographical
line courses or to prefer diagonal, horizontal or vertical edges.
Evaluations were run on an Intel Xeon E5649 machine with 12 cores (each with
2.53 GHz) and 96 GB of RAM. All our test datasets as well as the evaluation setup are
available online
3
. Our main results can be summarized as follows:
ILPs are too slow for practical use. Even for the simplest input dataset, the ILP took
minutes to optimize. For the Chicago dataset, it took 16 hours. For London and
New York, no feasible solution was found in under 24 hours.
There is no best input edge ordering for the approximate approach. Under the dif-
ferent edge orderings we tried for the approximate approach, we could nd no clear
winner.
Approximate approaches are well-suited for practice. For the datasets for which
the ILP yielded an optimal solution, the maximum percentage error of the approx-
imate approaches over all tested congurations was only 7.6%. Under visual in-
spection, we typically saw little to no dierence. All but the London and New York
dataset were octilinearized in under 1 second.
Sparse grids may spee d up the solution, but not always. While the sparse grids re-
duced the grid graph sizes by up the 82.7% on average, with only little quality loss
for the OHG-1 grid, and required less memory for optimization, some of the cor-
responding ILPs took much longer to optimize. For the approximate approach,
however, average solution times decreased by up to 28%.
Hexalinear and orthoradial maps can be quickly rendered using our approach.
For all but the London and New York datasets, our approach produced orthoradial
layouts in under 1.1 seconds, and hexalinear maps in under 3.3 seconds.
5.11.1 Integer Linear Program
We evaluated our ILP approach on all datasets with the degree-2 heuristic (LP-2), using
3 dierent grid cell sizes: D = 0.75 ·
¯
d, D = 1.0·
¯
d, and D = 1.25 ·
¯
d, where
¯
d is the average
distance between adjacent input nodes. Because the force-based distance optimization
penalty described in Section 5.5.8 is quadratic, we could not use it in our ILP approach
and therefore could not evaluate LP-2 with additional density penalty. The results for
3
https://github.com/ad-freiburg/octi-eval
220 Chapter 5. Schematic Transit Maps
Table 5.3: Dimensions, solution times t, and nal objective value Θ of our ILPs with the
degree-2 heuristic (LP-2) for grid cell sizes D = 0.75 ·
¯
d, D = 1.0 ·
¯
d and D = 1.25 ·
¯
d.
Solution times of — mean no initial feasible solution could be found in under 24 hours.
We then give the best bound of the objective function, if one was obtained by the solver.
D = 0. 75 ·
¯
d D = 1.0 ·
¯
d D = 1.25 ·
¯
d
rows×cols t Θ rows×cols t Θ rows×cols t Θ
FR 154.1k ×967.3k 5m 0.1k 90.4k ×550.1k 3m 0.1k 60.0k ×356.5k 1m 93.3
VN 249.2k ×1.6M 1h 0.2k 149.4k ×934.7k 48m 0.1k 109.5k ×672.7k 2h 0.1k
CG 667.8k ×4.4M 12h 0.3k 366.5k ×2.4M 16h 0.2k 244.7k ×1.6M — ≤0.2k
SD 976.5k ×6.5M 5h 0.4k 549.5k ×3.6M 47m 0.3k 358.1k ×2.3M 3h 0.2k
BE 907.0k ×6.0M 2h 0.3k 535.1k ×3.5M 58m 0.2k 344.4k ×2.2M — ≤0.2k
ST 1.1M×7.5M 2h 0.4k 658.0k ×4.3M 30m 0.3k 423.4k ×2.7M 2h 0.3k
LO 4.0M×26.7M — ≤0.7k 2.3M×15.6M — ≤0.6k 1.5M×9.9M — ≤0.3k
NY 15.5M×106.6M — — 8.6M×58.7M — — 5.5M×37.1M — ≤0.8k
the ILP sizes (given as number of rows × number of columns), the required optimization
time and the nal objective value for LP-2 are given in Table 5.3. We also evaluated the
ILP approach without the degree-2 heuristic in [21], but none of the datasets could be
optimized in under 24 hours.
All results in Table 5.3 were computed using gurobi 9.1.2 with default settings (the
number of threads was limited to 8 to make the memory consumption manageable), as
gurobi proved to be the ILP solver with the best overall performance. Table 5.4 gives
a comparison of the GLPK, COIN-OR CBC, and gurobi on all datasets. With GLPK, we
were not able to nd an optimal solution in under 24 hours for any dataset.
Each dataset was presolved with our approximate approach, and the approximate
solution provided to the solver. In [21] it was observed that this warm-start sped up
solution times by up to a factor of 2. Note that in the solver comparison in Table 5.4,
only GLPK and gurobi were warm-started with the approximate solution, as the COIN-
OR CBC solver library oers no straightforward way to provide a rst feasible solution.
For grid cell sizes of D = 0.75·
¯
d and D = 1.0·
¯
d, we were able to nd optimal solution
for all but the London New York dataset. For D = 1.0 ·
¯
d, these were typically found in
under 1 hour, but for D = 0.75 ·
¯
d and D = 1.25 ·
¯
d, optimization took signicantly longer.
For London and New York, we were not able to nd any feasible solution (the solution
provided by our approximate approach also had several constraint violations).
In general, the raw ILPs seem to be unsuitable for practical usage, as the optimization
takes impractically long, and requires much memory (see Table 5.11 for a comparison).
5.11 Experimental Evaluation 221
Table 5.4: Comparison of the performance of 3 dierent ILP solvers: the GNU linear
programming kit (GLPK), the COIN-OR CBC solver (CBC), and gurobi, evaluated on a
grid graph with cell size D = 1.0 ·
¯
d, and with the degree-2 heuristic enabled. Under t
we give the time to optimize the ILP, under mem we give the peak memory usage. A t
entry of — means that we were not able to nd an optimal solution in under 24 hours,
A mem entry of — means the optimization was terminated by the out-of-memory killer.
GLPK CBC gurobi
t mem t mem t mem
FR — — 8h 18.5 GB 3m 1.4 GB
VN — — — 31.4 GB 48m 2.9 GB
CG — — — 71.2 GB 16h 38.8 GB
SD — — — 94.7 GB 46m 10.5 GB
BE — — — 91.5 GB 58m 10.5 GB
ST — — — — 30m 11.1 GB
LO — — — — — —
NY — — — — — —
Moreover, the only solver that was able to nd optimal solutions for most of the datasets
in under 24 hours was the non-free gurobi solver. However, it seems promising to use the
ILPs to optimize the solutions found by our approximate approach for a very short time
(e.g. 0.5 seconds) to polish the approximate results. Further experiments are required
here, though. In particular, we also have to nd a way to incorporate the force-based
distance penalty (or a similar technique using linear penalties) into our ILPs.
5.11.2 Input Edge Orderings for Approximate Approach
Table 5.5 gives the nal objective functions of our approximate approach with the degree-
2 heuristic and without the local search phase using 6 dierent methods for ordering the
input edges as described in Section 5.5.1: by number of lines (num lines), by edge length
(length), by adjacent node degree (deg), by adjacent node line degree (ldeg), and using the
growth-based method with the node degree (gr-deg), or the node line degree (gr-ldeg).
None of the methods consistently outperformed the other methods. We also experi-
mented with reversing all 6 orderings or taking a random ordering, but the results were
distinctly inferior on all datasets.
For the evaluation of our approximate approach below, we always tried all 6 methods
for nding an initial drawing (this can be easily parallelized). The best solution was then
passed to the local search phase.
222 Chapter 5. Schematic Transit Maps
Table 5.5: Initial results of our heuristic approach without the local search phase using
6 dierent methods of ordering the input edges for rendering: by number of lines on
the edge (num lines), by the length of the edge in the input graph, shortest rst (length),
by adjacent node degree (deg), by adjacent node line degree (ldeg), and growth-based
approaches based on the node degree (gr-deg) and the line node degree (gr-ldeg). No
clear winner emerged.
num lines length deg ldeg gr-deg gr-ldeg
FR 126.4 128.9 126.1 118.4 120.8 118.9
VN 143.9 136.0 135.3 135.3 135.7 135.7
CG 3w
∞
244.6 16w
∞
1w
∞
251.5 269.7
SD 305.8 295.1 306.1 1w
∞
307.6 1w
∞
BE 3w
∞
259.0 262.5 270.7 4w
∞
4w
∞
ST 331.0 339.1 343.3 334.6 345.2 331.7
LO 88w
∞
67w
∞
72w
∞
85w
∞
43w
∞
22w
∞
NY 73w
∞
85w
∞
32w
∞
21w
∞
35w
∞
60w
∞
5.11.3 Quality and Speed of Approximate Approach
Our approximate approach was evaluated both without the degree-2 heuristic (A), and
with the degree-2 heuristic (A-2), and with additional density penalty (A-2+D). We fo-
cused on 2 aspects: the percentage error δ when compared to the optimal solution ob-
tained via an ILP, and the solution time. As mentioned above, we could not incorporate
the quadratic density penalty into our ILP. For A-2+D, we therefore could not measure
the approximation error. As we could not nd optimal solutions using our ILPs without
the degree-2 heuristic, we could also not measure the approximation error for A.
Table 5.6 gives the nal objective values, as well as the measured percentage errors
for our A-2 on all datasets, when compared to LP-2, again for 3 grid cell sizes: D = 0.75·
¯
d,
D = 1.0 ·
¯
d, and D = 1.25 ·
¯
d. The solution times and number of local search iterations
required for A, A-2, and A-2+D, can be seen in Table 5.7.
The solution quality of the approximate approach A-2 was very good, with average
percentage errors for datasets where an optimal solution was available between 1.2%
and 3.8% for dierent grid sizes D. Even for the highly complex London and New York
networks, there were only 1-2 constraint violations on a D = 0.75 ·
¯
d grid. However, for
larger grid sizes, this increased to up to 16 violations.
The A-2+D approach was able to produce high-quality drawings without topology
violations in under 1 second for all datasets except London and New York. Solution times
were even lower without the distance optimization between contracted stations (A-2).
5.11 Experimental Evaluation 223
Table 5.6: Final objective values of our ILP (LP-2) and our approx. approach (A-2), both
with deg-2 heuristic for grid cell sizes D = 0.75 ·
¯
d, D = 1.0 ·
¯
d and D = 1.25 ·
¯
d. The
percentage error is given as δ . Scores of — mean no solution could be found. If no
optimal LP solution was found in under 24 hours, we give the best bound. The average
percentage error only includes datasets for which an optimal solution was available.
D = 0.75 ·
¯
d D = 1.0 ·
¯
d D = 1.25 ·
¯
d
LP-2 A-2 δ LP-2 A-2 δ LP-2 A-2 δ
FR 144.4 147.1 1.9% 115.9 117.0 1.0% 93.3 94.7 1.5%
VN 163.7 164.0 0.2% 128.5 131.7 2.5% 110.8 112.4 1.4%
CG 286.4 290.7 1.5% 229.2 243.8 6.4% 198.0 213.1 7.6%
SD 360.8 363.6 0.8% 286.4 288.8 0.8% 248.0 251.4 1.4%
BE 301.3 304.4 1.0% 238.7 249.7 4.6% 208.0 216.6 4.1%
ST 383.3 391.9 2.2% 308.8 323.1 4.6% 267.2 285.1 6.7%
LO — 1w
∞
— — 16w
∞
— — 13w
∞
—
NY — 2w
∞
— — 7w
∞
— — 6w
∞
—
avg 1.2% 3.3% 3.8%
Table 5.7: Solution times for our linear program and our approximate approach, without
(LP, A) and with (LP-2, A-2) the degree-2 heuristic. For the approximate approaches, we
also evaluate the running time with an additional density penalty (A-2+D) and give the
number of local search iterations until convergence was reached.
LP A its. LP-2 A-2 its. A-2+D its.
FR — 0.3s 9 3m 75ms 1 0.2s 9
VN — 0.9s 15 48m 0.1s 5 0.2s 9
CG — 1.8s 14 16h 0.5s 4 0.7s 11
SD — 2.2s 13 47m 0.3s 6 0.6s 14
BE — 3.1s 24 58m 0.4s 4 0.5s 11
ST — 2.8s 18 30m 0.3s 5 0.7s 18
LO — 37.3s 72 — 4.4s 15 8.5s 28
NY — 2m 95 — 34.7s 40 33.7s 40
224 Chapter 5. Schematic Transit Maps
Table 5.8: Eects of various methods of base grid simplication on base grid graph
size, given as number of nodes |Ψ
0
| and number of edges |Ω
0
|. Under ‘red.‘ we give the
reduction of the number of edges when compared to the full extended grid
Full Convex Hull Quadtree OHG-1 OHG-2
|Ω
0
| |Ω
0
| red. |Ω
0
| red. |Ω
0
| red. |Ω
0
| red.
FR 15.8k 7.8k 50.6% 4.6k 71.1% 5.0k 68.7% 15.8k 0.0%
VN 23.7k 10.0k 57.9% 4.5k 80.8% 5.9k 75.3% 23.7k 0.0%
CG 43.9k 17.9k 59.2% 5.6k 87.3% 9.8k 77.6% 43.9k 0.0%
SD 63.0k 40.0k 36.5% 8.0k 87.2% 15.0k 76.1% 63.0k 0.0%
BE 45.6k 25.2k 44.7% 7.9k 82.7% 15.1k 66.9% 45.6k 0.0%
ST 52.7k 30.3k 42.5% 10.4k 80.3% 19.9k 62.2% 52.7k 0.0%
LO 78.4k 47.3k 39.7% 14.4k 81.6% 34.7k 55.8% 78.4k 0.0%
NY 221.2k 92.8k 58.0% 20.3k 90.8% 73.3k 66.9% 221.2k 0.0%
avg 48.6% 82.7% 68.7% 0.0%
5.11.4 Eect of Sparse Base Grids
For the evaluation of the sparse octilinear base grids, our interest was twofold: First,
we wanted the know the additional impact sparse base grids have on the solution qual-
ity. Second, we were interested in the gained solution time speedup, and memory con-
sumption reduction. As mentioned above and visible in Table 5.4, the ILP optimization
required signicant memory, making the latter particular important.
The eect of the sparse grid techniques on the dimensions of the underlying grid
graph can be seen in Table 5.8. Quadtree based grids reduced the number of edges by
82.7% on average, while the grids based on the convex hull of the input graph already
yielded an average reduction of 48.6%. A single-iteration octilinear Hanan grid (OHG-1)
gave an average reduction of 68.7%. For all datasets, the octilinear Hanan grid already
converged to the full grid after a second iteration (OHG-2), therefore resulting in no
reduction.
The impact on the solution quality depended on the type of sparse base grid. For the
convex hull based approach and octilinear Hanan grids (OHG-1), the average impact on
the solution quality of the ILP based approach was negligible: the average percentage
error was 1.4% for the convex hull based approach, and 1.9% for OHG-1 (Table 5.9). Fig-
ure 5.29 gives a visual example of a map rendered on a full grid, and various sparse grids.
For the quadtree based approach, the average percentage error was much higher at 9.5%,
and the ILP for the Chicago network became infeasible. Contrary to our initial hope, the
reduced ILP sizes only sometimes resulted in an optimization speedup. For example, al-
5.11 Experimental Evaluation 225
(1) (2)
(3) (4)
Figure 5.29: 2020 Freiburg tram network, octilinearized using an ILP on (1) a full grid,
(2) a convex hull based grid, (3) a quadtree based grid, and (4) an octilinear Hanan grid.
226 Chapter 5. Schematic Transit Maps
Table 5.9: Eects of various methods of base grid simplication on ILP sizes, solution
times and optimality. ILPs were optimized using gurobi.
Full Convex Hull Quadtree OHG-1 OHG-2
Θ t Θ δ t Θ δ t Θ δ t Θ δ t
FR 115.9 3m 116.8 0.8% 2m 122.1 5.4% 54.0s 117.8 1.6% 53.8s 115.9 0.0% 3m
VN 128.5 48m 129.4 0.7% 27m 145.6 13.3% 42m 130.9 1.9% 38m 128.5 0.0% 52m
CG 229.2 16h 238.8 4.2% 19h — — — 234.3 2.2% 6h 229.2 0.0% 13h
SD 286.4 47m 287.1 0.2% 41m 306.3 6.9% 52m 294.3 2.7% 54m 286.4 0.0% 1h
BE 238.7 58m 241.5 1.2% 5h 271.4 13.7% 13h 243.0 1.8% 6h 238.7 0.0% 1h
ST 308.8 30m 312.5 1.2% 4h 334.3 8.3% 7h 312.6 1.2% 28m 308.8 0.0% 37m
LO — — — — — — — — — — — — — —
NY — — — — — — — — — — — — — —
avg 1.4% 158.4% 9.5% 495.3% 1.9% 71.7% 0.0% 11.1%
though a single-iteration octilinear Hanan grid (OHG-1) sped up the solution times for
Freiburg, Vienna, and Chicago, the time required to optimize the Sydney dataset was
signicantly longer, and for the Berlin network, it went up from 58 minutes to 6 hours.
On average, the solution times hence increased by up to 495.3% for the quadtree based
approach. In hindsight, this was to be expected, though: the sparse base grids not only
make nding an initial feasible solution much harder, they are also much less regular
than a full grid.
The speedup was more pronounced for the approximate approach (Table 5.10), with
an average reduction of the required solution time by up to 28% for OHG-1. The approx-
imation error, when compared to the approximate approach on the full grid (not to the
optimal ILP solution!), was similar to the ILPs, and was also lowest for the convex hull
based approach, and highest for the quadtree based grid. Interestingly, the OHG-2 ap-
proach produced slightly better results than the full grid, although the grids had exactly
the same nodes and edges. The building process of the graphs, however, greatly dif-
fers, and the dierent orderings of the graph adjacency lists resulted in slightly dierent
solutions produced by the initial runs and the subsequent local search.
As expected, the memory consumption of both the ILP and the approximate approach
was often signicantly reduced (Table 5.11), but in some cases, the ILP solve consumed
more memory on the sparse grid than on the full grid (this was for example the case for
Sydney on an OHG-1 grid). We suspect a similar reason as for the increased optimization
times. Regarding memory consumption, solution time, and solution quality, a single-
iteration octilinear Hanan grid (OHG-1) seems to be a good compromise.
5.11 Experimental Evaluation 227
Table 5.10: Eects of various methods of base grid simplication on solution times and
optimality of our approximate approach.
Full Convex Hull Quadtree OHG-1 OHG-2
Θ t Θ δ t Θ δ t Θ δ t Θ δ t
FR 117.0 75ms 117.9 0.7% 52ms 133.9 14.5% 0.1s 120.2 2.7% 95ms 117.0 0.0% 82ms
VN 131.7 0.1s 132.9 0.9% 0.2s 152.2 15.5% 0.1s 134.1 1.8% 0.1s 129.4 -1.7% 0.1s
CG 243.8 0.5s 4w
∞
— 0.3s 2w
∞
— 0.3s 1w
∞
— 0.4s 242.9 -0.4% 0.5s
SD 288.8 0.3s 289.0 0.1% 0.3s 312.9 8.3% 0.3s 302.4 4.7% 0.3s 288.8 0.0% 0.4s
BE 249.7 0.4s 252.5 1.1% 0.3s 286.5 14.7% 0.3s 261.2 4.6% 0.2s 249.7 0.0% 0.4s
ST 323.1 0.3s 328.0 1.5% 0.3s 1w
∞
— 0.7s 325.9 0.9% 0.2s 318.1 -1.6% 0.3s
LO 16w
∞
4.4s 16w
∞
— 2.2s 61w
∞
— 1.8s 18w
∞
— 2.4s 18w
∞
— 5.4s
NY 7w
∞
34.7s 6w
∞
— 8.0s 45w
∞
— 9.0s 8w
∞
— 7.4s 7w
∞
— 28.3s
avg 0.9% -25.8% 13.3% -2.2% 2.9% -28.0% -0.6% 1.7%
Table 5.11: Memory consumption on sparse base grids of our ILP (LP-2) and our ap-
proximate approach (A-2), both with the degree-2 heuristic enabled.
Full Convex Hull Quadtree OHG-1
LP-2 A-2 LP-2 A-2 LP-2 A-2 LP-2 A-2
FR 1.4 GB 26.1 MB 795.3 MB 18.0 MB 896.1 MB 16.8 MB 985.2 MB 20.8 MB
VN 2.9 GB 33.3 MB 5.0 GB 21.1 MB 3.0 GB 17.4 MB 3.9 GB 23.2 MB
CG 38.8 GB 52.3 MB 20.5 GB 29.2 MB — 19.0 MB 8.0 GB 33.5 MB
SD 10.5 GB 68.8 MB 6.8 GB 48.2 MB 8.9 GB 23.4 MB 14.0 GB 44.0 MB
BE 10.5 GB 53.4 MB 5.3 GB 34.9 MB 10.5 GB 22.4 MB 8.6 GB 39.9 MB
ST 11.1 GB 59.7 MB 6.8 GB 39.2 MB 10.7 GB 26.3 MB 4.7 GB 48.3 MB
LO — 85.4 MB — 56.8 MB — 31.7 MB — 71.4 MB
NY — 220.1 MB — 100.7 MB — 43.2 MB — 149.6 MB
avg -17.7% -37.3% -6.4% -58.7% -14.7% -26.9%
228 Chapter 5. Schematic Transit Maps
Table 5.12: Solution times for our methods LP-2, A-2 and A-2+D on a hexalinear and a
pseudo-orthoradial base graph. The relative approximation error of our A-2 approach is
given as δ . If topology violoations occured, we do not give the approximation error, but
the number of violations in parentheses.
Hexalinear Pseudo-Orthoradial
LP-2 A-2 δ A-2+D LP-2 A-2 δ A-2+D
FR 2m 0.1s 10.1% 0.6s 41.7s 88ms 20.9% 0.2s
VN 8m 0.2s 6.7% 0.7s 30m 0.1s — (1) 0.1s
CG 18h 1.7s 5.1% 3.1s 3h 0.4s — (1) 0.8s
SD 30m 1.2s 7.4% 3.0s 13m 0.5s 5.3% 1.0s
BE 3h 1.3s 11.1% 3.3s 15h 0.8s — (2) 1.1s
ST 23m 1.2s 8.5% 3.0s 9m 0.7s — (2) 0.7s
LO — 7.1s — (10) 22.5s — 12.4s — (196) 9.1s
NY — 30.4s — (40) 2m — 26.0s — (180) 23.2s
5.11.5 Non-Octilinear Layout Experiments
Table 5.12 gives the running times and approximation errors of our A-2 and A-2+D ap-
proaches on a hexalinear and a pseudo-orthoradial grid (again, we could not evaluate
the approximation error of A-2+D). Although the solution times are signicantly higher
than for the octilinear setting, our A-2+D approach was still able to produce orthoradial
drawings for all but the London and New York datasets in under 1.1 seconds.
5.11.6 Visual Evaluation
Measuring the esthetic quality and the readability of schematic public transit maps (for
example, through a user study) is a research topic of its own and would have been beyond
the scope of this work. A recent survey by Wu et al. gives a thorough overview of
the challenges [151]. To still give proof that the maps rendered using our approach
are of high visual quality, various rendered maps of our testing datasets schematized
by our approaches, including renderings which consider obstacles and approximate the
geographical line courses, can be found at the end of this chapter. An additional visual
comparison between our approach and other approaches from the literature can be found
in Figure 5.4. All our schematic line graphs can also be inspected online
4
.
4
https://octi.informatik.uni-freiburg.de/
5.12 Conclusions and Future Work 229
Figure 5.30: Left: A historical map of the Chicago light rail system in which the high-
density Chicago Loop is enlarged using a sh-eye eect. Right: An ocial map of the
Chicago network in which the Loop is depicted as an enlarged cut-out.
5.12 Conclusions and Future Work
This chapter described a novel approach to generate schematic drawings of transit maps
by nding metro-map images of an input line graph in a special grid graph, with the
ability to consider obstacles or approximate the original geographical line courses. Moti-
vated by an NP-hardness proof for the general problem on octilinear grid graphs, we pre-
sented an ILP to nd such images. As this ILP proved to be impractically large and slow
to solve, we developed an approximation method that routes the input edges through
the grid graph in a predened order. To avoid stalling of this approach, we applied sev-
eral smart heuristics and relaxed hard constraints for topology preservation. To further
rene the metro-map images and to x hard constraint violations in the initial drawing,
we added a local-search step.
We demonstrated that our approach is able to nd high-quality octilinear metro-
map drawings fast (typically in under 1 second for small to intermediate-size networks)
and that it is also able to draw metro-maps following an ortholinear, hexalinear, or
(pseudo-)orthoradial layout. In an attempt to further speed up both the ILP solution
and our approximate approach, and to reduce the memory consumption, we presented
and evaluated several ideas for sparse base grids. We have also shown that the ability of
our approach to consider obstacles or approximate geographical line courses is not just
theoretical. Nevertheless, we regard our work in this chapter only as a rst step towards
fully automated schematic transit map generation using our approach. We do not yet
consider the space needed for line or station labels during our schematization process.
A posteriori labeling is possible, but will either fail if the schematization process did
230 Chapter 5. Schematic Transit Maps
Figure 5.31: Left: Line graph for the 2020 Freiburg tram network. Right: A (simpli-
ed) ocial schematic representation of the network. The network has been rotated by
around 40
◦
in counter-clockwise direction to improve the alignment of the main axes.
not leave enough space for a label, or may not allow for an optimal placement. Future
research has to address this problem.
We are also condent that more elaborate heuristics to prevent stalling of our ap-
proximate approach would help to reduce the number of topology violations for very
large networks. Also, we so far only used the constraint relaxation for our heuristic ap-
proach. It may also help our ILP to arrive at a feasible solution faster, and may enable
us to use a constraint-relaxed solution by our heuristic approach as a warm start for the
ILP solver. Besides this, we see the following directions for future work:
Stott et al. [133] already noted that when octilinear metro-map embeddings are gen-
erated by a local search over the station positions on a regular grid it may be benecial
to try not only moving individual nodes, but entire clusters of nodes to overcome a local
optimum. This idea directly translates to our approach, and using their clustered local
search may further improve our approximate maps.
Our approach so far respects the positions of the original input stations by punish-
ing station displacement. However, it is often desirable to enlarge certain parts of the
network in a metro-map drawing, for example to ensure the readability of high-density
areas by enlarging them into areas of lower density (for example, a city center enlarged
into the municipal area). Such transformations are common in real-world maps, where
the city-center is sometimes even depicted as an enlarged cut-out, or focused using a
sh-eye eect (see the examples of the Chicago loop in Figure 5.30). The latter tech-
nique heavily distorts the map and violates octilinearity, while the former makes it hard
to follow lines which cross the cut-out boundaries.
We see several ways to address this problem, they all transform the input line graph
prior to schematization: one idea is to use a quadtree to identify areas of high density,
which are then resized into empty neighboring cells. This transformation is completely
5.12 Conclusions and Future Work 231
local and would not aect other parts of the network. A second idea is to identify sta-
tion clusters in the input line graph using for example the k-means algorithm, and then
applying a focus transformation on the center of each cluster with a density above a
certain threshold. Last, we may use a classic spring embedder on the input line graph to
arrive at a uniform distance between adjacent stations.
Real-world input line graphs also often feature one or more major corridors which
are often drawn as straight vertical or horizontal segments in manually designed maps,
even though their course is not strictly following an east/west or north/south direction.
In eect, the manually designed maps are slightly rotated to enhance their appearance
(see Figure 5.31 for an example). We see two simple ways to do this automatically: either
we search for the rotation of the input graph for which the cumulated deviation of the
segments from the east/west or north/south axes is minimized (for example by checking
each rotation within a tolerable limit from +45
◦
to −45
◦
in 5 degree steps), or we octi-
linearize all of these rotated line graphs explicitly and pick the metro-map image of best
score.
In the remainder of this chapter, we show several example maps rendered by our
approach.
232 Chapter 5. Schematic Transit Maps
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Siebenhirten
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Karlsplatz
Volkstheater
Längenfeldgasse
Landstraße
Spittelau
Praterstern
Westbahnhof
Schottenring
Rathaus
Kagran
Kagraner Platz
Alaudagasse
Nestroyplatz
Rennbahnweg
Aderklaaer Straße
Tscherttegasse
Alte Donau
Gumpendorfer Straße
Niederhofstraße
Rochusgasse
Meidling Hauptstraße
Messe-Prater
Am Schöpfwerk
Perfektastraße
Zippererstraße
Kaisermühlen-VIC
Altes Landgut
Donauinsel
Vorgartenstraße
Taborstraße
Troststraße
Schlachthausgasse
Reumannplatz
Erdberg
Keplerplatz
Südtiroler Pl. - Hbf.
Gasometer
Taubstummengasse
Enkplatz
Währinger Str.-Volksoper
Kardinal-Nagl-Platz
Stadlau
Burggasse-Stadthalle
Kettenbrückengasse
Thaliastraße
Josefstädter Straße
Donaustadtbrücke
Pilgramgasse
Alser Straße
Donaumarina
Michelbeuern-AKH
Schottentor
Stadion
Margaretengürtel
Nußdorfer Straße
Schweglerstraße
Johnstraße
Großfeldsiedlung
Hütteldorfer Straße
Krieau
Kendlerstraße
Erlaaer Straße
Alterlaa
Hardeggasse
Meidling
Schönbrunn
Hietzing
Aspern Nord
Braunschweiggasse
Unter St. Veit
Neubaugasse
Ober St. Veit
Hausfeldstraße
Jägerstraße
Neulaa
Stadtpark
Dresdner Straße
Aspernstraße
Handelskai
Neue Donau
Donauspital
Friedensbrücke
Stubentor
Figure 5.32: Octilinear map of the 2020 Vienna subway network rendered by our ap-
proach.
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Ottakring
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Oberlaa
Leopoldau
Hütteldorf
Siebenhirten
Landstraße
Spittelau
Praterstern
Westbahnhof
Volkstheater
Schottenring
Längenfeldgasse
Stephansplatz
Schwedenplatz
Pilgramgasse
Neulaa
Messe-Prater
Krieau
Stadion
Donaumarina
Donaustadtbrücke
Stadlau
Hardeggasse
Donauspital
Aspernstraße
Hausfeldstraße
Aspern Nord
Kendlerstraße
Johnstraße
Nußdorfer Straße
Währinger Str.-Volksoper
Michelbeuern-AKH
Vorgartenstraße
Aderklaaer Straße
Großfeldsiedlung
Kagraner Platz
Kagran
Stadtpark
Alte Donau
Kaisermühlen-VIC
Donauinsel
Südtiroler Pl. - Hbf.
Keplerplatz
Reumannplatz
Troststraße
Altes Landgut
Alaudagasse
Rennbahnweg
Erlaaer Straße
Hietzing
Schönbrunn
Niederhofstraße
Meidling
Tscherttegasse
Nestroyplatz
Braunschweiggasse
Perfektastraße
Am Schöpfwerk
Alterlaa
Taborstraße
Schottentor
Rathaus
Zippererstraße
Josefstädter Straße
Thaliastraße
Schweglerstraße
Schlachthausgasse
Erdberg
Gasometer
Alser Straße
Enkplatz
Roßauer Lände
Neue Donau
Handelskai
Jägerstraße
Ober St. Veit
Unter St. Veit
Figure 5.33: Orthoradial map of the 2020 Vienna subway network rendered by our
approach.
5.12 Conclusions and Future Work 233
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1 3
4 2
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3
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2 3
3 5
Gundelfinger Str.
Moosweiher
Laßbergstraße
Innsbrucker Straße
Dorfstraße
Technische Fakultät
Bollerstaudenstraße
Munzinger Straße
Stadttheater
Bertoldsbrunnen
H.-von-Stephan-Straße
Rathaus im Stühlinger
Johanneskirche
Am Lindenwäldle
Robert-Koch-Straße
Komturplatz
Europaplatz
Hauptbahnhof
Eschholzstraße
Hornusstraße
Tullastraße
Glottertalstraße
Reutebachgasse
Holzmarkt
Runzmattenweg
Berggasse
Tennenbacher Straße
VAG-Zentrum
Hauptstraße
Eichstetter Straße
Okenstraße
Am Bischofskreuz
Betzenhauser Torplatz
Rennweg
Paduaallee
Moosgrund
Bugginger Straße
Diakoniekrankenhaus
Geschwister-Scholl-Platz
Schwabentorbrücke
Elsässer Straße
Berliner Allee
Musikhochschule
Klosterplatz
Wiesenweg
Reiterstraße
Pressehaus
Vauban-Mitte
Wonnhalde
Peter-Thumb-Straße
Weddigenstraße
Oberlinden
Emil-Gött-Straße
Holbeinstraße
Dorfbrunnen
Lorettostraße
Bissierstraße
Maria-von-Rudloff-Platz
Brauerei Ganter
Haslach Bad
Scherrerplatz
Maria-Hilf-Kirche
Hauptfriedhof
Hasemannstraße
Paula-Modersohn-Platz
Rohrgraben
Krozinger Straße
Römerhof
Alter Messplatz
Figure 5.34: Octilinear map of the 2020 Freiburg tram network rendered by our ap-
proach.
3
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Bollerstaudenstraße
Moosweiher
Munzinger Straße
Technische Fakultät
Laßbergstraße
Dorfstraße
Innsbrucker Straße
Stadttheater
Bertoldsbrunnen
Rathaus im Stühlinger
Johanneskirche
Am Lindenwäldle
Robert-Koch-Straße
Komturplatz
Eschholzstraße
Berggasse
Glottertalstraße
Runzmattenweg
Holzmarkt
Reutebachgasse
Tullastraße
Hornusstraße
Holbeinstraße
Weddigenstraße
Peter-Thumb-Straße
Paula-Modersohn-Platz
Vauban-Mitte
Lorettostraße
Musikhochschule
Wonnhalde
Wiesenweg
Klosterplatz
Freiburg Killianstraße
Oberlinden
Schwabentorbrücke
Brauerei Ganter
Maria-Hilf-Kirche
Alter Messplatz
Maria-von-Rudloff-Platz
Emil-Gött-Straße
Hasemannstraße
Römerhof
Hauptfriedhof
Rennweg
Eichstetter Straße
VAG-Zentrum
Moosgrund
Bissierstraße
Paduaallee
Rohrgraben
Betzenhauser Torplatz
Diakoniekrankenhaus
Am Bischofskreuz
Okenstraße
Hauptstraße
Tennenbacher Straße
Berliner Allee
Geschwister-Scholl-Platz
Reiterstraße
Pressehaus
Haslach Bad
Scherrerplatz
Dorfbrunnen
Krozinger Straße
Elsässer Straße
Figure 5.35: Orthoradial map of the 2020 Freiburg tram network rendered by our ap-
proach.
234 Chapter 5. Schematic Transit Maps
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Hauptbahnhof
Zwickauer Damm
Hönow
Innsbrucker Platz
Ruhleben
Alt-Tegel
Alt-Mariendorf
Hermannstr.
Pankow
Wittenau
Alexanderplatz
Hallesches Tor
Nollendorfplatz
Augsburger Str.
Gleisdreieck
Kottbusser Tor
Zoologischer Garten Bhf
HermannplatzFehrbelliner Platz
Bismarckstr.
Berliner Str.
Bayerischer Platz
Stadtmitte
Leopoldplatz
Möckernbrücke
Mehringdamm
Osloer Str.
Wittenbergplatz
Prinzenstr.
Görlitzer Bahnhof
Schlesisches Tor
Richard-Wagner-Platz
Oranienburger Tor
Britz-Süd
Naturkundemuseum
Johannisthaler Chaussee
Parchimer Allee
Lipschitzallee
Siemensdamm
Elsterwerdaer Platz
Wedding
Westphalweg
Jungfernheide Bhf
Mierendorffplatz
Halemweg
Jakob-Kaiser-Platz
Cottbusser Platz
Wutzkyallee
Rohrdamm
Paulsternstr.
Hellersdorf
Haselhorst
Zitadelle
Altstadt Spandau
Friedrichsfelde
Karl-Bonhoeffer-Nervenklinik
Podbielskiallee
Rehberge
Breitenbachplatz
Seestr.
Nauener Platz
Rüdesheimer Platz
Rathaus Reinickendorf
Heidelberger Platz
Afrikanische Str.
Lindauer Allee
Hansaplatz
Paracelsus-Bad
Turmstr.
Ernst-Reuter-Platz
Schönhauser Allee
Vinetastr.
Yorckstr.
Blaschkoallee
Louis-Lewin-Str.
Grenzallee
Neukölln
Wuhletal
Karl-Marx-Str.
Rathaus Neukölln
Schönleinstr.
Tierpark
Schwartzkopffstr.
Biesdorf-Süd
Borsigwerke
Holzhauser Str.
Otisstr.
Dahlem-Dorf
Scharnweberstr.
Friedrich-Wilhelm-Platz
Kurt-Schumacher-Platz
Rosenthaler Platz
Kleistpark
Paradestr.
Voltastr.
Onkel Toms Hütte
Deutsche Oper
Kaiserin-Augusta-Str.
Bernauer Str.
Samariterstr.
Weinmeisterstr.
Alt-Tempelhof
Weberwiese
Spittelmarkt
Oskar-Helene-Heim
Gneisenaustr.
Gesundbrunnen Bhf
Bundestag
Eisenacher Str.
Frankfurter Tor
Birkenstr.
Amrumer Str.
Senefelderplatz
Schillingstr.
Strausberger Platz
Blissestr.
Klosterstr.
Eberswalder Str.
Olympia-Stadion
Heinrich-Heine-Str.
Jannowitzbrücke
Lichtenberg
Konstanzer Str.
Wilmersdorfer Str.
Adenauerplatz
Platz der Luftbrücke
Ullsteinstr.
Neu-Westend
Kienberg
Theodor-Heuss-Platz
Kaulsdorf-Nord
Kaiserdamm
Südstern
Schloßstr.
Walther-Schreiber-Platz
Franz-Neumann-Platz
Residenzstr.
Leinestr.
Westhafen
Magdalenenstr.
Boddinstr.
Mohrenstr.
Frankfurter Allee
Freie Universität
Tempelhof
Figure 5.36: Octilinear map of the 2020 Berlin subway network rendered by our ap-
proach.
Figure 5.37: Orthoradial map of the 2020 Berlin subway network rendered by our ap-
proach.
5.12 Conclusions and Future Work 235
T8
T4
T2 T5
T5
T2
T4
T3
T2
T5
T1
T5 T1
T2 T5
T3
T9
T1
T2 T5
T3
T4
T8 T4
T4
T3
T1T1
T5 T1
T1
T3
T6
T3
T1
T1
T9
T1
T5 T1
T6
T1 T2
T3
T5 T1
T1 T2
T5 T1
T9 T1
T4
T5
T1
T1
T7
T5 T1
T2 T5
T8
T9
T1
T2
T3
T8
T
4
T4
T8
T1
T5
T1
T2
T8
T4
T2 T5
T8
T8
T8 T8 T8
T4
T
9
T
1
T1
T4
T4
T9
T8
T9
T8
T9
T8
T3
T2
T9
T5 T1
Richmond
Leppington
Cronulla
Emu Plains
Bondi Junction
Macarthur
Waterfall
Berowra
Olympic Park
Carlingford
Central
St James
Glenfield
Lidcombe
Wolli Creek
Town Hall
Macdonaldtown
Wynyard
Homebus
Harris Park
Canley Vale
Sydenham
Blacktown
Clyde
Granville
Regents Park
Sutherland
Hornsby
Redfern
Petersham
Strathfield
Cabramatta
Museu
Warwick Farm
Burwood
Circular Quay
Lewisham
Stanmore
Summer Hill
Newtow
Erskineville
Ashfield
St Peters
Croydon
Parramatta
Liverpool
Vineyard
Quakers Hill
North Sydney
Schofields
Seven Hills
Toongabbie
St Leonards
Milsons Point
Riverstone
Waverton
Marayong
Wollstonecraft
Pendle Hill
Clarendon
Yennora
Killara
Lindfield
Roseville
Guildford
Chatswood
Fairfield
Auburn
Tempe
Westmea
Flemington
Edmondson Park
Artarmon
Casula
Wentworthville
East Richmond
Mulgrave
Windsor
Merrylands
Gordon
Warrawee
Concord West
Arncliffe
North Strathfield
Banksia
Waitara
Rockdale
Kogarah
Wahroonga
Carlton
Rhodes
Turramurra
Allawah
Pymble
Marrickville
Hurstville
Loftus
Penshurst
Dulwich Hill
Mortdale
Jannali
Cheltenham
Mount Kuring-Gai
Normanhurst
Martin Place
Thornleigh
Caringbah
Miranda
Woolooware
Pennant Hills
Gyme
Beecroft
Kirrawee
Birrong
Berala
Epping
Green Square
Eastwood
Mascot
Denistone
Domestic Airport
West Ryde
Meadowban
International Airport
Chester Hill
St Marys
Kingswood
Sefton
Telopea
East Hills
Panania
Lakemba
Dundas
Campbelltown
Wiley Park
Mount Druitt
Ingleburn
Leumeah
Punchbowl
Macquarie Fields
Doonside
Rooty Hill
Rydalmere
Bankstown
Leightonfield
Camellia
Yagoona
Villawood
Rosehill
Carramar
Beverly Hills
Oatley
Com
Hurlstone Park
Heathcote
Asquith
Mount Colah
Turrella
Edgecliff
Kings Cross
Bardwell Park
Bexley North
Kingsgrove
Engadine
Narwee
Holsworthy
Canterbury
Campsie
Riverwood
Padstow
Revesby
Werrington
Penrith
Belmore
Minto
Figure 5.38: Octilinear map of the 2020 Sydney light rail network rendered by our ap-
proach.
T9
T1
T2
T5
T8
T1
T4
T2
T5
T8
T1
T2
T6
T
8
T
4
T4
T
4
T
2
T
8
T
3
T1 T2
T8
T
2
T
5
T8
T
1
T
2
T8
T1
T
2
T
5
T1
T3
T1
T5
T
4
T8
T
6
T2 T5
T3
T9
T5
T
2
T
9
T1
T
1
T
2
T
9
T
2
T
5
T
3
T1
T2
T9
T8
T4
T
8
T4
T9
T
2
T
8
T
3
T
8
T
3
T4
T3
T1
T5
T
1
T
2
T
5
T1 T5
T1
T5
T8
T4
T
4
T1
T5
T1
T2
T3
T8
T
9
T4
T1 T9
T1
T2
T1 T2
T1
T8
T6
T3
T
3
T1
T3
T1
T5
T9
T
2
T5
T
3
T
8
T
4
T1
T5
T1
T
7
T
4
T9
T8
T1 T5
T1 T2
T9
T1 T9
T
4
T4
T1 T9
T
9
T
9
T3
T
8
T9
T3
T1 T9
T1
T1 T9
T1 T9
T4
T1
T5
T1
T8
T4
T
4
T1 T5
T1
T5
T6
T1
T5
T
3
T4
T2
T5
T6
T
6
T1 T9
T3
T3
T4
T
1
T2
T5
T8
T1
T5
T8
T
9
T3
T2
T5
T
4
T3
T1
T1
T8
T4
T1
T2
T9
T1
T4
T1
T2 T5 T3
T
4
T8
T8
T3
T
2
T
5
T1 T9
T1
T3
T
9
T9
T8
T1
T5
T4
T
9
T8
T4
T4
T8
T1 T9
T1
T5
T1 T9T1 T9
T3
T
1
T
2
T
9
T1
T5
T3
T
4
T
8
T4
T4
T3
Richmond
Leppington
Berowra
Waterfall
Cronulla
Emu Plains
Carlingford
Olympic Park
Bondi Junction
Macarthur
St James
Glenfield
Lidcombe
Wolli Creek
Macdonaldtown
Town Hall
Wynyard
Homebush
Sydenham
Harris Park
Canley Vale
Granville
Blacktown
Clyde
Sutherland
Hornsby
Regents Park
Redfern
Croydon
Ashfield
Burwood
Lewisham
St Peters
Cabramatta
Petersham
Stanmore
Warwick Farm
Strathfield
Museum
Circular Quay
Summer Hill
Parramatta
Liverpool
Merrylands
Mulgrave
Edmondson Park
East Richmond
Milsons Point
St Leonards
North Sydney
Westmead
Seven Hills
Wollstonecraft
Casula
Tempe
Fairfield
Guildford
Windsor
Artarmon
Waverton
Pendle Hill
Flemington
Riverstone
Schofields
Quakers Hill
Toongabbie
Marayong
Yennora
Clarendon
Wentworthville
Chatswood
Killara
Vineyard
Lindfield
Roseville
Auburn
Gordon
Penrith
East Hills
Mount Colah
Mortdale
Leumeah
Panania
Warrawee
Kingswood
Campsie
Pymble
Chester Hill
Eastwood
Holsworthy
Narwee
Loftus
Heathcote
Bexley North
Rockdale
Bankstown
Epping
Rhodes
Canterbury
Revesby
Gymea
Miranda
Kingsgrove
Banksia
Denistone
Caringbah
Mount Kuring-Gai
Thornleigh
Concord West
Carramar
Mount Druitt
Marrickville
International Airport
Domestic Airport
Kogarah
Rooty Hill
Oatley
Como
Turramurra
Kirrawee
Dundas
Normanhurst
Sefton
Werrington
Yagoona
Telopea
Macquarie Fields
Wahroonga
Asquith
Pennant Hills
North Strathfield
Ingleburn
Belmore
Punchbowl
Woolooware
Carlton
Campbelltown
Minto
Beecroft
Lakemba
Jannali
Birrong
Turrella
Arncliffe
Leightonfield
Dulwich Hill
Camellia
Rydalmere
Penshurst
Berala
Rosehill
Edgecliff
Beverly Hills
St Marys
Padstow
Engadine
Hurstville
Waitara
Riverwood
Wiley Park
Cheltenham
Villawood
Meadowbank
West Ryde
Allawah
Doonside
Bardwell Park
Figure 5.39: Orthoradial map of the 2020 Sydney light rail network rendered by our
approach.
236 Chapter 5. Schematic Transit Maps
U8 U7
U15
U8 U7
U12
U14
U8
U7
U14
U14
U8
U7
U1
U14
U9
U14
U13
U5
U11 U19
U14
U14
U6
U1
U15
U7
U5
U1
U3
U1
U13
U1
2
U4
U
6
U7
U7
U7
U3
U6
U2
U2
U
1
U1
U13
U2
Vaihingen
Nellingen
NeckarPark
Botnang
Hed
elfingen
Lutherkirche
Hallschlag
Plieningen
Gerlingen
Mönchfeld
Hölderlinplatz
Stammheim
Leinfelden Bahnhof
Neckargröningen Remseck
Fasanenhof Schelmenwasen
Neugereut
Killesberg
Charlottenplatz
Hauptbahnhof
Mercedesstr.
Berliner Platz
Pragsattel
Wilhelmsplatz
Löwentor
Rosensteinbrücke
Staatsgalerie
Stöckach
Möhringen Bf
Pragfriedhof
Olgaeck
Österreichischer Platz
Eckartshaldenweg
Wasenstr.
Bergfriedhof
Wangener-/Landhausstr.
Ebitzweg
Zuffenhausen Kelterplatz
Neckartor
Vaihinger str.
Mineralbäder
Peregrinastr.
Weinsteige
Stadtmitte
Metzstr.
Bopser
Sonnenberg
Riedsee
Dobelstr.
Degerloch Albstr.
Degerloch
SSB-Zentrum
Jurastr.
Löwentorbrücke
Wallgraben
Silberwald
Sillenbuch
Schemppstr.
Bockelstr.-Heumaden
Heumaden
Technische Akademie
Lindpaintnerstr.
Arndt-/Spittastr.
Ruit
Beethovenstr.
Sportpark Feuerbach
Föhrich
Waldau
Wilhelm-Geiger-Platz
Bihlplatz
Wangen Marktplatz
Feuerbach
Hedelfinger str.
Schloss-/Johannesstr.
Südheimer Platz
Friedrichswahl
Hohensteinstr.
Vogelsang
Zinsholz
Salamanderweg
Marienplatz
Im Degen
Maybachstr.
Borsigstr.
Brendle (Großmarkt)
Scharnhauser Park
Inselstr.
Bergheimer Hof
Wolfbusch
Rohrer Weg
Scharnhauser Park Kreuzbrunnen
Rastatter str.
Parksiedlung
Schwab-/Bebelstr.
Herderplatz
Möhringen Freibad
Weilimdorf Löwen-Markt
Millöckerstr.
Landauer str.
Uff-Kirchhof
Eltinger str.
Pfostenwäldle-Feuerbach
Karl-Olga-Krankenhaus
Giebel
Untertürkheim Bahnhof
Vogelrain
Steinhaldenfeld
Eugensplatz
Schwabenlandhalle
Plieninger str.
Obere Ziegelei
Daimlerplatz
Fasanenhof
Landhaus
Salzäcker
Sigmaringer str.
Kienbachstr.
Bubenbad
Blick
Nürnberger str.
Eszet
EnBW City
Schlotterbeckstr.
Europaplatz
Hauptfriedhof
Payerstr.
Unteraichen
Augsburger Platz
Rosensteinpark
Kursaal
Antwerpener str.
Heutingsheimer str.
Beskidenstr.
Höhenstr.
Leinfelden Frank
Esslinger str.
Gnesener str.
Schlachthof
Engelboldstr.
Aldingen Mühle, Remseck am Neckar
Kaltental
Mühlhausen
Waldeck
Auwiesen
Max-Eyth-See
Hofen
Wagrainäcker
Elbestr.
Glockenstr.
Freibergstr.
Wilhelma
Münster Rathaus
Münster Viadukt
Mühlsteg
Kraftwerk Münster
Riethmüllerhaus
Aldingen Hornbach, Remseck am Neckar
Züricher str.
Salzwiesenstr.
Kirchtalstr.
Wimpfener str.
Stelle
Ostheim Leo-Vetter-Bad
Korntaler str.
Heidehofstr.
Schozacher str.
Fürfelder str.
Tapachstr.
Himmelsleiter
Suttnerstr.
Gerlingen Siedlung
Gaisburg
Freiberg
Breitwiesen
Stafflenbergstr.
Ostendplatz
Geroksruhe
Milchhof
Mittnachtstr.
Russische Kirche
Schillerplatz
Nordbahnhof
Fauststr.
Raitelsberg
Vaihingen Viadukt
Aldingen Brückenstr.
Figure 5.40: Octilinear map of the 2015 Stuttgart light rail network rendered by our
approach.
5.12 Conclusions and Future Work 237
U2
U
9
U
1
3
U2
U2
U8
U7
U
1
5
U8
U7
U1
U
8
U
7
U
1
5
U8
U7
U
8
U
7
U15
U
1
3
U1
U1
U1
U1
U14
U14
U14
U14
U
1
4
U9
U13
U3
U
1
3
U
7
U15
U
9
U
5
U12
U
9
U1
U1
U12
U6
U5
U6
U
3
U
3
U3
U6
U
7
U7
U15
U15
Vaihingen
Hedelfingen
NeckarPark (Stadion)
Nellingen
Botnang
Mönchfeld
Killesberg
Hallschlag
Hölderlinplatz
Lutherkirche
Stammheim
Leinfelden Bahnhof
Fasanenhof Schelmenwasen
Neckargröningen Remseck
Plieningen
Neugereut
Gerlingen
Hauptbf (Arnulf-Klett-Platz)
Mercedesstraße
Pragsattel
Wilhelmsplatz
Löwentor
Staatsgalerie
Stöckach
Olgaeck
Pragfriedhof
Möhringen Bf
Österreichischer Platz
Eckartshaldenweg
Wasenstraße
Ebitzweg
Rosensteinbrücke
Zuffenhausen Kelterplatz
Bergfriedhof
Wangener-/Landhausstraße
Vaihinger Straße
Mineralbäder
Stadtmitte
Weinsteige
Degerloch
Degerloch Albstraße
Peregrinastraße
Sonnenberg
Riedsee
Dobelstraße
Bopser
Metzstraße
Silberwald
Löwentorbrücke
Bockelstraße-Heumaden
Jurastraße
Wallgraben
Schemppstraße
Sillenbuch
SSB-Zentrum
Heumaden
Sportpark Feuerbach
Salamanderweg
Eltinger Straße
Pfostenwäldle-Feuerbach
Landauer Straße
Technische Akademie
Möhringen Freibad
Weilimdorf Löwen-Markt
Parksiedlung
Zinsholz
Wolfbusch
Bergheimer Hof
Rastatter Straße
Föhrich
Wilhelm-Geiger-Platz
Feuerbach
Maybachstraße
Ruit
Waldau
Scharnhauser Park Kreuzbrunnen
Sieglestraße
Beethovenstraße
Vogelsang
Lindpaintnerstraße
Scharnhauser Park
Wangen Marktplatz
Schwab-/Bebelstraße
Schloss-/Johannesstraße
Herderplatz
Im Degen
Brendle (Großmarkt)
Inselstraße
Karl-Olga-Krankenhaus
Hedelfinger Straße
Südheimer Platz
Rohrer Weg
Hohensteinstraße
Friedrichswahl
Borsigstraße
Bihlplatz
Millöckerstraße
Untertürkheim Bahnhof
Giebel
Vogelrain
Gaisburg
Unteraichen
Ostheim Leo-Vetter-Bad
Leinfelden Frank
Ostendplatz
Kienbachstraße
Gnesener Straße
Steinhaldenfeld
Heutingsheimer Straße
Landhaus
Salzäcker
Plieninger Straße
Sigmaringer Straße
Raitelsberg
Korntaler Straße
Wimpfener Straße
Salzwiesenstraße
Freiberg
Suttnerstraße
Himmelsleiter
Tapachstraße
Fürfelder Straße
Schozacher Straße
Kirchtalstraße
Gerlingen Siedlung
Rosensteinpark
Breitwiesen
Hofen
Nürnberger Straße
Augsburger Platz
Schlotterbeckstraße
Eszet
Blick
Mühlsteg
Kraftwerk Münster
Münster Viadukt
Münster Rathaus
Freibergstraße
Elbestraße
Wagrainäcker
Max-Eyth-See
Wilhelma
Auwiesen
Mühlhausen
Aldingen Hornbach, Remseck am Neckar
Aldingen Mühle, Remseck am Neckar
Aldingen Brückenstraße
Waldeck
Schillerplatz
Fauststraße
Vaihingen Viadukt
Engelboldstraße
Schwabenlandhalle
Daimlerplatz
Fasanenhof
Europaplatz
EnBW City
Stelle
Geroksruhe
Kaltental
Riethmüllerhaus
Nordbahnhof
Milchhof
Hauptfriedhof
Esslinger Straße
Höhenstraße
Beskidenstraße
Antwerpener Straße
Kursaal
Payerstraße
Bubenbad
Obere Ziegelei
Heidehofstraße
Schlachthof
Eugensplatz
Stafflenbergstraße
Figure 5.41: Orthoradial map of the 2015 Stuttgart light rail network rendered by our
approach.
238 Chapter 5. Schematic Transit Maps
U3
U2
U4
U6
U3
U6
U4
U6
U6
U6
U1
U3
U4
U1
U6
U2U2U2
U6
U4
U2
U6
U6
U2
U6
U4
U4
U2
U4U4
U4
U3
U4
U3
U2U2
U4
U2U2
U1
U2
U1
U1
U4
U2
U2
U2
U1
U2
U1
U1
U1
U1
U1
U1
U3
U6
U6
U6
U4
U6
U6
U6
U6
U6
U4
U3
U4
U4
U3
U1
U1
U1
U1
U1
U1
U2
U1
U3
U3
U3
U3
U3
Oberlaa
Siebenhirten
Heiligenstadt
Hütteldorf
Simmering
Seestadt
Ottakring
Leopoldau
Floridsdorf
Spittelau
Praterstern
Schwedenplatz
Schottenring
Westbahnhof
Längenfeldgasse
Volkstheater
Landstraße
Josefstädter Straße
Johnstraße
Kardinal-Nagl-Platz
Kettenbrückengasse
Nußdorfer Straße
Schweglerstraße
Alser Straße
Thaliastraße
Stadtpark
Handelskai
Dresdner Straße
Neue Donau
Roßauer Lände
Alte Donau
Schlachthausgasse
Erdberg
Gasometer
Zippererstraße
Enkplatz
Vorgartenstraße
Schottentor
Kaisermühlen-VIC
Donauinsel
Kagran
Hütteldorfer Straße
Kagraner Platz
Aderklaaer Straße
Rennbahnweg
Rochusgasse
Rathaus
Margaretengürtel
Pilgramgasse
Großfeldsiedlung
Kendlerstraße
Braunschweiggasse
Donauspital
Niederhofstraße
Meidling
Hietzing
Schönbrunn
Aspern Nord
Unter St. Veit
Hardeggasse
Tscherttegasse
Perfektastraße
Donaumarina
Donaustadtbrücke
Stadlau
Alterlaa
Am Schöpfwerk
Erlaaer Straße
Messe-Prater
Taubstummengasse
Südtiroler Pl. - Hbf.
Keplerplatz
Reumannplatz
Troststraße
Altes Landgut
Alaudagasse
Neubaugasse
Jägerstraße
Neulaa
Taborstraße
Krieau
Stadion
Aspernstraße
Hausfeldstraße
Ober St. Veit
Figure 5.42: Hexalinear map of the 2020 Vienna subway network rendered by our ap-
proach.
U1
U2
U8
U7
U6
U7
U2
U1
U3
U9
U1
U5
U6
U5
U8
U6
U5
U6
U8
U6
U6
U3
U2
U4
U7
U9
U6
U5U5
U7
U9
U1 U3
U8
U2 U5
U6
U4
U5
U5
U3
U5U5
U55
U9
U6
U
5
5
b
au
U9
U6
U6
U3
U3
U
9
U5
U9
U3
U6
U5
U7
U5U2
U7
U3
U3
U6
U2 U2
U9
U9
U2
U6
U5
U8
U6
U7
U3
U7
U9
U5U5
U7
U1 U3
U3
U8
U8
U8
U
U7
U8
U7 U7 U7 U7
U7 U7
U7
U
7
U4
U7
U8
U8
U8
U7
U7
U7
U7
U7
U7U7
U7
U7
U7
U8
U8
U2U2U2U2U2
U7
U7
U7
U4
U5U5 U5
U3
U3
U8
U6
U6
U8 U8
U8
U8
U9
U2
U6
U6
U6
U6
U6
U6
U2
U2
U2
U9
U2
U9
U9
U2
U7
Warschauer Str.
Ruhleben
Rathaus Spandau
Rathaus Steglitz
Uhlandstr.
Innsbrucker Platz
Wittenau
Zwickauer Damm
Hermannstr.
Krumme Lanke
Alt-Tegel
Berlin Hauptbahnhof
Alt-Mariendorf
Hönow
Pankow
Kottbusser Tor
Fehrbelliner Platz
Leopoldplatz
Hermannplatz
Bismarckstr.
Mehringdamm
Osloer Str.
Schlesisches Tor
Prinzenstr.
Kurfürstenstr.
Görlitzer Bahnhof
Jannowitzbrücke
Leinestr.
Sophie-Charlotte-Platz
Mierendorffplatz
Lipschitzallee
Wutzkyallee
Kaiserdamm
Boddinstr.
Haselhorst
Jungfernheide Bhf
Richard-Wagner-Platz
Moritzplatz
Johannisthaler Chaussee
Rathaus Neukölln
Karl-Marx-Str.
Neukölln
Blaschkoallee
Grenzallee
Britz-Süd
Parchimer Allee
Eberswalder Str.
Vinetastr.
Kurt-Schumacher-Platz
Afrikanische Str.
Otisstr.
Scharnweberstr.
Holzhauser Str.
Borsigwerke
Senefelderplatz
Schönhauser Allee
Nauener Platz
Schloßstr.
Walther-Schreiber-Platz
Friedrich-Wilhelm-Platz
Bundesplatz
Rüdesheimer Platz
Neu-Westend
Olympia-Stadion
Wilmersdorfer Str.
Adenauerplatz
Tierpark
Friedrichsfelde
Kienberg (Gärten der Welt)
Berlin, Theodor-Heuss-Platz
Heidelberger Platz
Franz-Neumann-Platz
Seestr.
Rehberge
Lindauer Allee
Paracelsus-Bad
Karl-Bonhoeffer-Nervenklinik
Rathaus Reinickendorf
Hellersdorf
Elsterwerdaer Platz
Deutsche Oper
Lichtenberg
Kaiserin-Augusta-Str.
Frankfurter Allee
Westphalweg
Samariterstr.
Frankfurter Tor
Brandenburger Tor
Tempelhof
Dahlem-Dorf
Podbielskiallee
Mohrenstr.
Weberwiese
Louis-Lewin-Str.
Cottbusser Platz
Ullsteinstr.
Hansaplatz
Wedding
Naturkundemuseum
Breitenbachplatz
Gneisenaustr.
Westhafen
Altstadt Spandau
Kaulsdorf-Nord
Wuhletal
Onkel Toms Hütte
Rosenthaler Platz
Voltastr.
Blissestr.
Pankstr.
Zitadelle
Kleistpark
Paulsternstr.
Rohrdamm
Siemensdamm
Halemweg
Turmstr.
Biesdorf-Süd
Friedrichstr. Bhf
Bundestag
Freie Universität (Thielplatz)
Alt-Tempelhof
Strausberger Platz
Jakob-Kaiser-Platz
Magdalenenstr.
Oskar-Helene-Heim
Paradestr.
Amrumer Str.
Birkenstr.
Residenzstr.
Figure 5.43: Hexalinear map of the 2020 Berlin subway network rendered by our ap-
proach.
5.12 Conclusions and Future Work 239
Figure 5.44: Octilinear map of the 2020 London subway network rendered by our ap-
proach.
240 Chapter 5. Schematic Transit Maps
2
4 2
2
1
1
4
3
1
3
1
1
5
1
5
1
5
4
3
3
2
4
2
2
3
4 2
5
5
2
3
Gundelfinger Str.
Innsbrucker Straße
Bollerstaudenstraße
Technische Fakultät
Dorfstraße
Laßbergstraße
Munzinger Straße
Moosweiher
Sta
d
tth
ea
ter
Bertoldsbrunnen
H.-
v
on-
Step
han
-
Straße
Rathaus im Stühlinger
Rober
t
-
Koc
h
-
S
t
r
aße
Am Lindenwäldle
J
o
h
a
nn
e
sk
i
rc
h
e
Komturplatz
Europaplatz
H
au
p
tb
a
h
n
h
o
f
Eschholzstraße
Glottertalstraße
R
un
z
m
att
en
w
eg
Tullastraße
Reutebachgasse
Berggasse
Hornusstraße
Ho
l
zm
ark
t
Wiesenweg
Klosterplatz
Reiterstraße
Paula-Modersohn-Platz
Fahnenbergplatz
Bissierstraße
V
aub
a
n-
M
it
t
e
Rohrgraben
Maria-von-Rudloff-Platz
Lorettostraße
Bugginger Straße
Holbeinstraße
VAG-Zentrum
Geschwister-Scholl-Platz
Wonnhalde
M
u
s
i
k
h
o
c
hs
c
hu
le
Am Bischofskreuz
Elsässer Straße
Berliner Allee
Brauerei Ganter
Maria-Hilf-Kirche
A
l
ter
M
e
s
s
p
l
a
t
z
Moosgrund
E
m
i
l
-G
ö
t
t
-S
t
raße
Hasemannstraße
Römerhof
Hauptfriedhof
Rennweg
Eichstetter Straße
Peter-Thumb-Straße
Sc
h
wa
b
ento
r
b
r
ücke
Diakoniekrankenhaus
K
r
oz
ing
er
Straße
Paduaallee
Dorfbrunnen
Betzenhauser Torplatz
Scherrerplatz
Okenstraße
Hauptstraße
Weddigenstraße
Haslach Bad
Tennenbacher Straße
Figure 5.45: Octilinear map of the 2020 Freiburg tram network approximating geo-
graphical line courses, overlayed over satellite imagery.
Figure 5.46: Ortholinear map of the 2020 Freiburg tram network, respecting obstacles
of parks, forest, and mountains.
Chapter 6
Rendering and Labeling
We close the algorithmic part of this work with a short and practical description of how
the line graphs generated by the methods presented so far can be rendered into a map.
This chapter thus addresses the following problem:
Problem 6 (Line Graph Rendering). Given a line graph G = (V , E, L , L) and a line order-
ing for G, render G into a map.
The input line graph G may be either geographically correct, or schematic. We as-
sume that the line graphs have been prepared in a way resembling the approach de-
scribed in Chapter 3. In particular, this means that edge segments within a threshold
distance
ˆ
d have been merged.
6.1 Overview
As mentioned in the introduction of this work, we chose our denitions of a line graph
and a line ordering in such a way that they will allow for easy rendering. Our basic
approach, depicted in Figure 6.1, is hence straightforward: Given an input line graph
G (1), we rst render all lines L(e) passing through an edge e in the order given by the
respective line ordering as osetted polylines (2). Recall that although G is undirected,
we always give the line orderings w.r.t. to an adjacent node to ensure that the render-
ing order is well-dened. Observe that adjacent line segments will inevitably overlap
at intersection nodes. In a process we call node expansion, we therefore rst free the
intersection nodes by these overlapping parts (3) and reconstruct the intersection by
connecting lines in adjacent edge segments with Bézier curves (4). Then the stations are
rendered on top (5). In a nal and optional step, labels are added for the lines and the
stations (6). Each of these steps will be further explained below.
241
242 Chapter 6. Rendering and Labeling
(1)
Station A
Station B
Station C
Station D
(A, B, C)
(A, D)
(B, C)
(D)
(2) (3)
(4) (5) (6)
Station B
Station A
Station C
Station D
B
C
B
C
D
A
A
D
Figure 6.1: Overview over our rendering approach. Given an input line graph (1), render
the ordered lines as osetted polylines (2), expand nodes (3), reconstruct node connec-
tions with Bézier curves (4), render station markers (5), and label the map (6).
6.2 Osetted Line Rendering
Let e be an input edge e = {u, v} which is embedded in R
2
be a polyline P(e), and let
(l
1
, ··· , l
n
), l
i
∈ L(e)be the line ordering for e. W.l.o.g., we assume that the line ordering is
given w.r.t. tov. We then render each l
i
as a polyline of width w which is perpendicularly
osetted from the supporting polyline P(e) by −w
|
L(e)
|
/2+w
(
i − 1
)
. If P(e)is a straight
line segment, this is simple. Osetting an arbitrary polyline, however, is nontrivial, and
known as (open) polygon osetting, or polygon buering, a process which is closely
related to the problem of constructing a straight skeleton of a polygon [4]. A straight
skeleton of a polygon P can be thought as being composed of lines tracking the nodes
of P when P is shrunk.
As polygon buering is a common requirement in many elds, many approaches
have been proposed in the literature (see [34] for an overview). The general diculty is
that when the polygon sides are osetted individually in a perpendicular fashion, they
will often overlap and introduce self-intersections.
For this work, we used a relatively simple approach for osetting the supporting
polyline, as depicted in Figure 6.2: let P = (s
1
, ··· , s
n
) be an open polygon, dened by
its straight line segments s
i
= (p
i
, p
i+1
), that should be osetted by some distance d. We
iterate over the segments in the order given by P and perpendicularly oset each s
i
by d,
resulting in a new osetted segment s
0
i
. The lines (not the line segments!) described by
s
0
i
and the previously inserted segment s
0
i−1
intersect at some point q, which we add as a
new support point for P
0
(for the rst end last segment, we simply keep the rst (or last)
6.2 Osetted Line Rendering 243
(1)
𝑃
𝑠
1
𝑠
2
𝑠
3
𝑞
𝑠
0
1
𝑠
0
2
𝑠
0
3
𝑑
(2)
𝑃
𝑃
0
Figure 6.2: Osetting a polyline (an open polygon) by a distance d. (1) We iterate over
the polyline segments s
1
, . . . , s
n
, oset each s
i
by d and determine the intersection q of
the lines described by the osetted segment s
0
i
and the last osetted segment s
0
i−1
. q is
added as a new support point to the osetted polygon P
0
(2).
(1)
𝑃
𝑠
1
𝑠
2
𝑞
𝑠
0
1
𝑠
0
2
(2)
𝑃
𝑃
0
Figure 6.3: (1) If the intersection point q of the lines described by two osetted line
segments s
0
1
and s
0
2
is too far away from s
0
1
or s
0
2
, we cap the resulting spike as depicted.
(2) The resulting osetted polyline.
point of the osetted segment). Note that if s
0
i
and s
0
i−1
are colinear, the intersection point
q is also not well-dened. We avoid such cases by rst simplifying the input polyline
using the Douglas-Peucker algorithm.
If two adjacent segments s
1
and s
2
meet at an acute angle, the line intersection point
q of their osetted segments s
0
1
and s
0
2
may be far away from s
0
1
and s
0
2
, as shown in
Figure 6.3. This produces undesirable spikes in the nal map rendering. To mitigate this
eect, we clip the corresponding segments if the distance is above the oset distance d
and introduce a new connecting segment, as shown in Figure 6.3.2.
The approach described so far only considers local intersections between adjacent
segments, and topological errors may still be introduced by segments which do not
appear consecutively. Figure 6.4 gives an example. We clean such errors in a post-
processing step in which we detect self-intersections and prune the corresponding loop
from P (Figure 6.4.3).
244 Chapter 6. Rendering and Labeling
(1)
P
s
1
s
2
s
3
(2)
P
i
P
0
(3)
P
P
0
Figure 6.4: (1) Our osetting approach may fail to catch intersections between osetted
line segments which were not originally adjacent in P. This will results in topologically
incorrect self-intersections in the osetted polyline (2). We apply a ltering step in which
we detect such self-intersections and delete the corresponding loop (3).
𝑝
𝑞
𝑝
0
𝑞
0
𝐵
Figure 6.5: A cubic Bézier curve B with its
four control points p, p
0
, q
0
, and q.
6.3 Inner Node Line Connections
The packed osetted lines of each edge s{u, v} can now be thought of describing a poly-
gon. We call the polygon segments that touchu or v the node fronts of e for the respective
nodes, and denote the node front of e at u as F
eu
. To make room for the inner node con-
nections at intersection nodes, we now move for each node v all adjacent node fronts
simultaneously inward with xed step sizes until a stopping criterion is met (we use a
distance threshold between the node fronts). Figure 6.1.3 gives an example. As men-
tioned above, we then connect corresponding lines with a Bézier curve. For our work,
we used cubic Bézier curves, as dened in the following [58].
Denition 6.1 (Cubic Bézier Curve). Given four control points p, p
0
, q, and q
0
in R
2
, a
cubic Bézier curve is dened by a polynomial function B : [0, 1] 7→ R
2
:
B(t) = (1 − t)
3
p + 3(1 −t)
2
tp
0
+ 3(1 −t)t
2
q
0
+ t
3
q. (6.1)
p is the starting point, q the endpoint of B. Intuitively, p − p
0
and q − q
0
can be
thought of dening vectors which describe the direction of the curve at p and q, and the
magnitudes |p −p
0
| and |q −q
0
| describe how much the curve is “forced” to follow that
6.3 Inner Node Line Connections 245
(1)
𝑝
𝑞
𝑖
𝑝
0
𝑞
0
𝐹
𝑓 𝑢
𝐹
𝑒𝑢
𝑓
𝑒
𝐴
𝐴
𝑡
𝑓 𝑢𝐴
𝑡
𝑒𝑢𝐴
𝑑
(2)
𝑝
𝑞
𝑝
0
𝑞
0
𝑑
(3)
𝑝
𝑞
𝑝
0
𝑞
0
𝑐
𝑖
Figure 6.6: (1) Inner node line connections for a line A between two node fronts F
eu
and
F
f u
drawn as cubic Bézier curves between p and q with control points p
0
and q
0
positioned
on line segments t
euA
and t
f uA
(both of length d = |p − q|) following the direction of A
on the last 5 meters before reaching the node point geometries. The position of p
0
and
q
0
on t
euA
and t
f uA
is based on the lengths |p − i | and |q − i |. (2) If t
euA
and t
f uA
do not
intersect, the position is based directly on d. (3) Bézier curve approximating a circular
segment with center c.
direction. B is tangential to the lines pp
0
and qq
0
. Figure 6.5 gives an example.
Let now A be a transit line passing through a node u from adjacent edge e to adjacent
edge f , p the position where A arrives at the node front F
eu
, and q the position where A
arrives at the node front F
f u
(Figure 6.6.1). We rst determine unit length vectors d
euA
and d
f uA
which describe the direction of transit line A at the respective node front by
sampling the corresponding polylines of A close to F
eu
and F
f u
(for our experiments, we
used a distance to the node fronts of 5 meters). Using these direction vectors, we then
draw two line segments t
f uA
and t
euA
starting at p and q, respectively, and going into the
direction d
euA
and d
f uA
. As a length of t
f uA
and t
euA
we use the distance d = |p −q|.
There are now two cases: if t
f uA
and t
euA
intersect at some point i, we average the
length of the segments pi and qi, that is we compute a δ = (|p − i| + |q − i |)/2. The
control point p
0
is then placed at p + d
euA
· k · δ, and the control point q
0
is placed at
q + d
f uA
· k · δ (Figure 6.6.1). We set k = 4/3(
(2) − 1) ≈ 0.55228474. If t
f uA
and t
euA
do not intersect, they are parallel, or close to parallel, and we simply place the control
points at p
0
= p + d
euA
·k ·d, and q
0
= q + d
f uA
·k ·d (Figure 6.6.2).
A particularly nice property of this setup using the magic number k = 0.55228474
is the following: if the intersection point i exists, and if |p − i| = |q − i|, then δ =
|p −i| = |q −i|, and the corresponding cubic Bézier Curve B approximates a circular arc
connecting p and q [125] (the approximation error is negligible in practice [73]). This
makes for a particularly clean look of schematic networks. Figure 6.6.3 gives an example
for the orthogonal case where the circle center is at c = q −δd
euA
. However, it is easy to
see that d
euA
and d
f uA
do not have to be orthogonal.
246 Chapter 6. Rendering and Labeling
Figure 6.7: Dierent station marker styles in (from left to right) the ocial transit maps
of Zurich, London, Freiburg, and Sydney. For another styling example, also compare
Figure 4.41.
(1) (2)
Figure 6.8: (1) Excerpt from a rendered map of the Freiburg tram network, without
station markers. Expanded node fronts are shown in green. (2) Stations rendered by
taking the convex hull of the node front geometries, with node fronts overlayed in red.
6.4 Station Rendering
For the station rendering, we have to consider two cases: either a station is served by all
adjacent lines, or some lines pass through u without a stop.
6.4.1 Fully Served Stations
Let u ∈ S be a station node with corresponding node fronts F
1
, ··· , F
n
. To render a
station which is served by all adjacent lines, we simply take the padded convex hull of
{F
1
, ··· , F
n
} and use the resulting polygon as a station marker, as shown in Figure 6.8
(for stations of degree 2 that only serve a single line we use a simpler approach and
render them as round markers).
While taking the convex hull produces compact station markers, real-world public
transportation maps often feature rectangular station nodes. To achieve this look, we
6.4 Station Rendering 247
(1) (2)
Figure 6.9: (1) Station marker rendered using the convex hull of the node fronts (2)
Station marker rendered using the oriented bounding box of the node fronts.
(1) (2) (3)
Figure 6.10: (1) Station marker using the oriented bounding box of the unexpanded node
fronts, producing a tighter look. (2) First expanding the node fronts allows for a clean
visualization of the line courses, for example by making the station node transparent.
instead compute the oriented bounding box of the {F
1
, ··· , F
n
}. Figure 6.9 compares
both approaches. Note that the node front expansion is not strictly necessary for station
nodes if they are rendered in this fashion. In fact, leaving the node fronts unexpanded
for station nodes will produce tighter station markers, as shown in Figure 6.10.1. How-
ever, also expanding station nodes allows to interactively hide the station markers, or to
render stations in a way which makes it possible to follow the line courses through the
stations. Figure 6.10.2 and 3 gives two examples. Also recall Figure 1.3, where stations
were drawn as transparent polygons.
(1) (2)
Figure 6.11: Station markers rendered directly on the node fronts for stations not served
by all adjacent lines. (1) The pink line does not serve the center stop. (2) Both the pink
line and the orange line do not serve the stop.
248 Chapter 6. Rendering and Labeling
U15
U15
U
1
5
U15
U9
U14 U1
1
U4
U
2
U
9
U2
U9
U
4
U
4
Hölderlinplatz
Charlottenplatz
Berliner Platz (Hohe Straße)
Hauptbf (Arnulf-Klett-Platz)
Stöckach
Staatsgalerie
Neckartor
Stadtbibliothek
Schlossplatz
Olgaeck
Stadtmitte
Friedrichsbau
/Spit
Schloss-/Johannesstraße
Schwab-/Beb
Rosenberg-/Seidenstraße
Russische Kirche
Heidehofstraße
Eugensplatz
Stafflenbergstraße
Bubenbad
Figure 6.12: Excerpt from
the Stuttgart light rail network
map, labeled using our prelim-
inary approach. Terminus la-
bels are rendered in bold text.
6.4.2 Partially Served Stations
For stations that are only served by some adjacent lines, we take a simpler approach and
render the station markers directly onto the node fronts. Figure 6.11 gives two examples.
Let u be a station node adjacent to lines L = {l
1
, ··· , l
n
}, and let N ⊂ L be a set of lines
which do not serve u. We then iterate over each node front and add a station marker
onto the incoming position of an unmarked line l
i
< L, and mark l
i
as processed. If
two station markers are next to each other on the same node front, we merge them. An
obvious extension to this method is to optimize the distribution of the station markers in
such a way that it creates as few markers as possible. For example, in Figure 6.11.1, the
method may place the station marker for the green and orange line onto the lower edge,
leaving a single station marker for the red line to be placed on the left or right edge.
This style of station markers may also be used on stations served by all lines - con-
sider for example the ocial maps of London and Sydney shown in Figure 6.7.
6.5 Labeling
Map labeling (sometimes called automatic label placement) is a large research topic of
its own and can be considered an optimization problem: place a set of labels (each for
a geometry on the map) in such a way that they do not overlap (conict), and that as
many labels as possible are visible. For surveys or literature, see for example [136] or the
map-labeling bibliography maintained by Alexander Wol [150]. It can be considered a
geometric maximum independent set problem, in which two labels are adjacent in the
graph-theoretical sense if they overlap. The problem is NP-hard [61]. Typically, there is
a set of position candidates for each label (for example, placed to the left, right, top, or
bottom). A simple approach which has found its way into some web mapping libraries
1,2
is to use a local search, typically simulated annealing [39], to approximate an optimal
1
https://github.com/migurski/Dymo
2
https://github.com/tinker10/D3-Labeler
6.5 Labeling 249
Bertoldsbrunnen
Stadttheater
0
2
6
3
1
3
6
2
Figure 6.13: Placement candidates for
a station label. At each octilinear ori-
entation, weighted by a score, we place
a candidate polygon (the bounding box
of the text label) at 3 text baseline po-
sitions. The conict-free candidate of
lowest score is chosen.
labeling. The problem of labeling dynamic maps which change over time (for example
by panning or zooming) or even feature moving objects has also found considerable
research interest in recent years [72].
For transit maps, there are typically two classes of labels: station labels and line
labels. For the latter, three approaches are predominant in manually designed maps: (1)
the line labels are placed on the lines (see for example the Freiburg map in Figure 6.7),
(2) the line labels are placed at their terminus stations, and (3) the line labels are placed
in their respective color along the edge segments. In case (1), no collisions can occur. In
case (2), the line label is part of the station label. We therefore only considered case (3)
in our work so far. In the context of schematic transit maps, map labeling is often part
of the schematization process, as the schematization may inuence the possible label
positions, and vice versa. As already hinted at in Chapter 5, map labeling was not the
focus of this work. We nevertheless implemented a preliminary labeling method.
The approach we chose is a straightforward greedy approach which places the labels
on the map consecutively: we rst order the station nodes (in descending order) by their
degree to ensure that larger stations are labeled rst. For each label, we then rotate a
polygon corresponding to the label’s bounding box by multiples of 45
◦
and check for each
of these octilinear orientations whether a collision occurs (Figure 6.13). If so, we discard
the position. If not, we assign each rotation a score, given here in clockwise direction
starting at 0
◦
: (6, 2, 0, 3, 6, 4, 1, 5, 6). The scores favor positions in which the line labels
are easy to read, and the reading order starts at the station node. We pick the position
with the lowest score. For the line labels, we try positions along the corresponding
edge and weight collision-free placements by their distance to the midpoint of the edge
geometry. If no collision-free placement is available, we do not display the label. While
the resulting labeling is certainly not perfect, we consider them satisfactory. Further
work is required here, though. Figure 6.14 shows a labeled geographically accurate map
of the Stuttgart light rail network, created using our approach described so far.
250 Chapter 6. Rendering and Labeling
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Vaihingen
NeckarPark
Botnang
Hedelfingen
Nellingen, Ostfildern
Stammheim
Hölderlinplatz
Fasanenhof Schelmenwasen
Gerlingen
Killesberg
Hallschlag
Leinfelden Bahnhof, Leinfelden-Echterdingen
Remseck
Mönchfeld
Plieningen
Neugereut
Lutherkirche, Fellbach
Berliner Platz
Löwentor
Rosensteinbrücke
Wasenstr.
Zuffenhausen Kelterplatz
Hauptbahnhof
Stöckach
Staatsgalerie
Neckartor
Möhringen Bf
Stadtbibliothek
Vaihinger Str.
Olgaeck
Metzstr.
Bopser
Dobelstr.
Stadtmitte
Pragsattel
Rathaus
Sonnenberg
Degerloch Albstr.
Degerloch
Peregrinastr.
Weinsteige
Eckartshaldenweg
Riedsee
Mineralbäder
Jurastr.
Friedrichsbau
Löwentorbrücke
Wallgraben
SSB-Zentrum
Silberwald
Sillenbuch
Schemppstr.
Bockelstr.-Heumaden
Heumaden
Technische Akademie, Ostfildern-Nellingen
Herderplatz
Lindpaintnerstr.
Österreichischer Platz
Marienplatz
Arndt-/Spittastr.
Millöckerstr.
Eltinger Str.
Inselstr.
Im Degen
Brendle
Wangener-/Landhausstr.
Bergfriedhof
Schloss-/Johannesstr.
Erwin-Schoettle-Platz
Vogelsang
Hedelfinger Str.
Wangen Marktplatz
Südheimer Platz
Bihlplatz
Beethovenstr.
Scharnhauser Park Kreuzbrunnen, Ostfildern
Zinsholz, Ostfildern-Ruit
Ruit, Ostfildern
Wilhelm-Geiger-Platz
Sieglestr.
Borsigstr.
Friedrichswahl
Hohensteinstr.
Feuerbach
Föhrich
Maybachstr.
Möhringen Freibad
Rohrer Weg
Sportpark Feuerbach
Pfostenwäldle-Feuerbach
Landauer Str.
Weilimdorf Löwen-Markt
Rastatter Str.
Wolfbusch
Salamanderweg
Parksiedlung, Ostfildern
Scharnhauser Park, Ostfildern
Uff-Kirchhof
Waldau
Karl-Olga-Krankenhaus
Bergheimer Hof
Giebel
Heslach Vogelrain
Untertürkheim Bahnhof
Daimlerplatz
Kaltental
Engelboldstr.
Vaihingen Viadukt
Schillerplatz-Vaihingen
Gnesener Str.
Riethmüllerhaus
Kursaal
EnBW City
Wimpfener Str.
Heutingsheimer Str.
Korntaler Str.
Mittnachtstr.
Nordbahnhof
Schlachthof
Raitelsberg
Europaplatz
Schozacher Str.
Milchhof
Ostendplatz
Höh
enstr., Fellbach
Ostheim Leo-Vetter-Bad
Pragfriedhof
Leinfelden Frank, Leinfelden-Echterdingen
Rosenberg-/Seidenstr.
Kirchtalstr.
Salzwiesenstr.
Stelle
Geroksruhe
Schwabenlandhalle, Fellbach
Esslinger Str., Fellbach
Waldeck
Augsburger Platz
Beskidenstr.
Antwerpener Str.
Payerstr.
Russische Kirche
Fauststr.
Steinhaldenfeld
Hauptfriedhof
Obere Ziegelei
Auwiesen
Wilhelma
Mühlsteg
Kraftwerk Münster
Münster Viadukt
Münster Rathaus
Freibergstr.
Elbestr.
Wagrainäcker
Max-Eyth-See
Hofen
Glockenstr.
Mühlhausen
Aldingen Hornbach
Aldingen Mühle
Aldingen Brückenstr.
Heidehofstr.
Gerlingen Siedlung
Breitwiesen, Gerlingen
Tapachstr.
Stafflenbergstr.
Sigmaringer Str.
Plieninger Str.
Salzäcker
Fasanenhof
Nürnberger Str.
Bubenbad
Freiberg
Suttnerstr.
Himmelsleiter
Fürfelder Str.
Leinfelden Unteraichen, Leinfelden-Echterdingen
Züricher Str.
Schlotterbeckstr.
Eszet
Blick
Kienbachstr.
Ebitzweg
Gaisburg
Landhaus
Figure 6.14: Labeled and geographically accurate map of the 2015 Stuttgart light rail
network.
Chapter 7
Software
This chapter provides on overview over (and short manuals for) the software developed
in the course of this work. The order in which the software is presented matches the
order in which the implemented methods were presented in Chapters 2 to 6. We took
the pipeline metaphor seriously: the last 4 tools presented here (topo, loom, octi, and
transitmap) can be plugged together using Unix pipes, which makes it straightforward
to extend our approach. All of our developed tools are open source and available online.
7.1 pfaedle - Map-Matching GTFS Data
The map-matching approach for public transit data presented in Chapter 2 has been
implemented in a C++ tool called pfaedle
1
. Our tool expects an OpenStreetMap dataset
and a schedule dataset given in the GTFS format as input, and outputs a fully map-
matched GTFS dataset with a shape for each trip. For example, the following command
will create shapes for a GTFS dataset in freiburg-gtfs from OSM data in freiburg.osm
and write it to the (default) folder gtfs-out:
$ pfaedle -x freiburg.osm freiburg-gtfs
It is highly congurable. In particular, the process of building the network graph for
dierent methods of transportation (MOT) can be controlled via extensive lter rules,
and adding additional MOTs simply amounts to adding a new section in the congu-
ration le. Other features include the ability to output ltered OSM les that contain
exactly the part of the data needed to map-match one or more input GTFS feeds, or a
matching Overpass API
2
query to obtain this data. Instead of outputting the shaped
GTFS feeds, pfaedle may also output the network graph as a GeoJSON le with trips
map-matched onto it.
1
https://github.com/ad-freiburg/pfaedle/
2
https://wiki.openstreetmap.org/wiki/Overpass_API
251
252 Chapter 7. Software
Figure 7.1: Station similarity
classication GUI for statsimi.
Users may click on a map to add
station position markers, which
can then be labeled freely. The
classication result is displayed
on top.
pfaedle was published in 2018. Based on feedback we received from various organi-
zations and transit companies, it has since become a standard tool for the generation of
missing shape information for GTFS feeds. It is also used in our transit visualization tool
TRAVIC
3
[17, 18] to weekly map-match many of the GTFS input dataset (among them
the nationwide schedule datasets for Germany, Switzerland, Sweden, and Denmark).
7.2 statsimi - Similarity Classication of Public Tran-
sit Stations
As part of our eort to nd the best station similarity classier, we developed statsimi
4
, a
Python tool that is able to extract station similarity ground truth data from OSM datasets
and can be used to nd station similarities either for individual stations via a built-in
HTTP interface, or in bulk. It can be used to train and evaluate all the classiers described
in Section 2.8. For example, a classication model for Germany can be trained on an OSM
le germany.osm.bz2 as follows:
$ statsimi model --model_out model.mod --train germany.osm.bz2
A classication server using this model and listening on port 9999 can then for ex-
ample be started like this:
$ statsimi http --model model.mod --http_port 9999
It serves a classication API at /api and a classication GUI at / (Figure 7.1).
An additional feature is to write x les for OSM data. These are tab-separated
les which contain for each station node or platform in a given input OSM le a list
a condence scores of how correct individual labels and the grouping of the nodes (via
3
https://travic.app/
4
https://github.com/ad-freiburg/statsimi
7.3 staty - Quality Assessment of OSM Station Tagging 253
Figure 7.2: Our web tool staty for assessing the quality of public transit station tagging
in OSM, based on x les writting by statsimi.
OSM relations) are, based on similarity classication results. For condence scores below
a certain threshold, labeling and grouping suggestions are provided, which are obtained
by re-clustering the station nodes based on the similarity scores, and translating the
dierence between the original grouping, and the re-clustered grouping into a set of
suggestion rules. See [16] for details. A x le for Germany using the model trained
above can be generated as follows:
$ statsimi fix --model model.mod --fix_out de.fix --test germany.osm.bz2
7.3 staty - Quality Assessment of OSM Station Tagging
We have build a weekly updated web application called staty
5
which visualizes x les
generated by statsimi on a map. The app covers Western Europe, the United States, and
Australia / New Zealand and features a heat-map overview to quickly see where errors in
the OSM data have been found, and where suggestions are provided. Individual grouping
suggestions are displayed on higher zoom levels, as shown in Figure 7.2.
7.4 topo - Map Construction for Line Graphs
For rendering transit maps, we have implemented a C++ software suite called LOOM
(Line Ordering Optimized Maps). The line graph construction approach from Chapter 3
has been implemented in a tool called topo, which expects an arbitrary (possibly over-
lapping) line graph as a GeoJSON le and outputs a line graph which is ready for line
ordering optimization, schematization, or direct rendering as a transit map. For convert-
ing GTFS schedule datasets into a preliminary line graph which can be fed to topo, a tool
5
https://staty.cs.uni-freiburg.de/
254 Chapter 7. Software
gtfs2graph is additionally provided.
All tools in LOOM (except gtfs2graph and transitmap) expect a line graph on stdin,
and write the result to stdout. For example, to extract a free line graph for the Freiburg
tram network from the ocial GTFS data (placed in a folder freiburg), the following
command suces:
$ gtfs2graph -m tram freiburg | topo > freiburg.json
7.5 loom - Line-Ordering Optimization
All methods from Chapter 4 for nding the optimal line-ordering for a given input line
graph have been implemented in the loom tool, which expects a line graph in stdin and
writes the ordering-optimized line graph to stdout. A rich set of command line options
may be used to specify for example the weights for line crossings and line separations,
and the method used for optimization. By default, it internally uses the improved ILP
approach with prior line graph simplication, solved with gurobi (if gurobi is not avail-
able, loom automatically falls back to rst CBC, then GLPK). The following command
nds the optimal line ordering for the Freiburg graph generated above:
$ loom < freiburg.json > freiburg.opt.json
For individual input datasets, one of the heuristic approaches described in Section 4.6
may be faster and sucient. For example, the greedy lookahead heuristic can be used as
follows:
$ loom -m greedy-lookahead < freiburg.json > freiburg.opt.json
7.6 octi - Fast Schematization of Line Graphs
The line graph schematization methods from Chapter 5 have been implemented in a tool
called octi. It expects a line graph on stdin and outputs a schematic version of the line
graph to stdout. For example, the line-ordering optimized line graph of the Freiburg
tram network from above can be octilinearized as follows:
$ octi < freiburg.opt.json > freiburg.octi.json
The same tool can also create ortholinear, hexalinear, or orthoradial drawings:
$ octi -b porthoradial < freiburg.opt.json > freiburg.ortho.json
A GeoJSON le of obstacles (rivers, lakes, mountain ranges etc.) may be provided via
the --obstacles parameter:
7.7 transitmap - Transit Map Rendering Engine 255
B
C
A
E
C
B
E
B
A
Main St
East St
South St
West St
Figure 7.3: Rendering of a simple input
line graph (with optimized line order-
ing) with labeling, directed line mark-
ers, and CSS styling (dashed line), gen-
erated by our tool transitmap.
$ octi --obstacles blackforest.json < freiburg.opt.json > freiburg.octi.json
To approximate the original input line courses to some degree p, it oers the param-
eter --enf-geo-pen=<p>.
7.7 transitmap - Transit Map Rendering Engine
Finally, the rendering and labeling approach described in Chapter 6 has been imple-
mented in a tool called transitmap, which expects a line graph from stdin, and writes
an SVG le to stdout. For example, the demo line graph example2.json provided with
the LOOM software suite can be rendered as follows into an image map.svg:
$ transitmap < example2.json > map.svg
transitmap supports dierent line widths, dierent line spacing, and can render di-
rection markers for directed lines (see Figure 7.3). As the output is an SVG graphic, it
is easily customizable using CSS rules (see the dashed line in Figure 7.3). These rules
can be embedded into the line graph le. Line graphs may also be provided in the DOT
language, although this is still experimental.
transitmap is the last component in our toolchain used for generating (schematic)
transit maps from raw schedule data. For example, an octilinear schematic transit map
of the Freiburg tram network can be elegantly created from the raw GTFS schedule data
in folder freiburg with the following command:
$ gtfs2graph -m tram freiburg | topo | octi | loom | transitmap > map.svg
As mentioned above, the strengths of this pipeline approach are that it can be easily
extended, and individual components replaced. For example, a drop-in replacement for
octi which schematizes the input line graph using the approach based on Bézier curves
by Fink et al. [58] was implemented as a student project
6
(see Figure 7.4).
6
https://ad-blog.informatik.uni-freiburg.de/post/circular-transit-maps/
Chapter 8
Conclusions
This thesis described a full end-to-end approach to automatically render transit maps
from schedule data. To achieve this, we identied six key problems from the elds of
map-matching, similarity classication, map-construction, combinatorial optimization,
graph drawing, and map rendering. We have developed algorithmic approaches to all
of these problems and presented reference implementations in publicly available open-
source tools.
Our map-matching method for nding missing shapes in schedule data adapted the
state-of-the-art approach of using a hidden Markov model and also considered public-
transit specic metadata from the transportation network. We demonstrated how both
intra-hop and inter-hop turn restrictions can be naturally considered by routing on the
edge-to-node dual of the transportation network graph. To better incorporate station
metadata, we evaluated several methods for station similarity classication, among them
a machine-learning based approach. A comprehensive evaluation showed the high qual-
ity of our approach: for example, for the Seattle and Vienna schedule data, we achieved
accuracy scores of nearly 100%. Our method also seems to be more robust against im-
precise input data than the standard method used in many routing and map-matching
libraries. A major challenge was nding the missing shapes in reasonable time for very
large input datasets. With a carefully designed set of speedup techniques, we achieved
a running time of 8.5 minutes for the entire national schedule data of Germany. This did
not include the parsing of the transportation network from OpenStreetMap (OSM) data,
though, as our reference implementation pfaedle so far only supports the raw OSM XML
format. As this is a major practical limitation, further work in pfaedle should extend the
support to arbitrary OSM formats, and also oer an option to serialize the transportation
network graph to disk, for fast later use. Our method so far also only utilizes the tem-
poral information contained in the schedule data to a limited extent. While arrival and
departure times in real-world schedule datasets are usually imprecise when compared to
the actual travel time of the vehicle, they may nevertheless provide valuable information
to prune unrealistic vehicle routes (for example, routes that are much too short). An ad-
257
258 Chapter 8. Conclusions
ditional challenge for future research is thus to better consider this temporal data. One
idea that seems promising is to generate more transition candidate paths (for example,
by using methods for calculating alternative routes) and base the transition probabilities
also on the scheduled travel time.
To prepare the shaped schedule data for later rendering, we developed a map con-
struction method to transform an arbitrary line graph (a graph where edges are labeled
with sets of transit lines, and nodes may be stations) into a free line graph, which is ready
for rendering without line overlaps and with stations clustered using our station similar-
ity classication approach. The input line graph may be directly created from schedule
data, or even from a transportation network like contained in OSM. This method was an
iteration on previous approaches. In this new approach, we did not track station nodes
or line turn restrictions through the constructions process, but added them onto the -
nal constructed graph. Although our method already produces satisfying results, we
still consider it preliminary. During the evaluation, it emerged that clustering stations
into single nodes was a bad idea - in some cases, there is no placement in the free line
graph so that the clustered station node is served by all lines which originally served
the clustered stops. The implementation of our approach so far uses preliminary line
graphs obtained from (map-matched) schedule data. It would be highly interesting to
extract such line graphs on a national, or even global level from OSM data, and test our
construction approach on such datasets. This would enable us to create a global transit
map from OSM data using the pipeline described in this work.
The combinatorial optimization problem of nding optimal permutations of the line
labels for the edges in the free line graph was particularly challenging. Here, the goal
was to produce as few line crossings in the nal map as possible. We formulated the
Metro Line Node Crossing Minimization problem (MLNCM), better suited for render-
ing than the classic Metro Line Crossing Minimization problem (MLCM), and developed
variants in which the crossings are weighted (MLNCM-W), or which additionally con-
sider the novel concept of line separations (MLNCM-S, MLNCM-WS), We proved the
NP-hardness of MLNCM (and hence MLNCM-W and MLNCM-WS). This motivated the
development of two integer linear programs (ILP) and several approximate approaches
to tackle this problem. Using a sophisticated set of simplication rules which implicitly
compute optimal partial line orderings and transform an input line graph into (typi-
cally many) components, we were able to optimize MLNCM-WS instances in under 0.6
seconds even for very large networks. We see potential in developing additional simpli-
cation rules. In particular, we outlined a whole class of new untangling rules, namely
induced untangling rules.
To generate schematic transit maps, we developed a novel approach for a classic
problem from graph drawing: metro map drawing. Our approach is based on nding
metro map images on special octilinear grid graphs and can also consider obstacles (e.g.
lakes or mountain ranges) or approximate the geographical course of the input lines. We
259
developed an ILP as well as a fast approximate approach to nd the optimal images. For
intermediate to large input networks (Chicago, Sydney, Berlin, Stuttgart), our approach
was able to nd high-quality octilinear maps in under 700 ms. Even for the highly com-
plex London subway network, we arrived at a solution in 8.5 seconds, albeit with several
constraint violations. We investigated the eect of sparse base grids on the quality and
solution time and found that our idea of using an octilinear Hanan grid constructed
from the input graph as a base grid greatly reduced the base grid size with only minimal
impact on the solution quality. The impact on the solution time, however, was below
our expectation. Regarding maps following other layouts, we showed that our approach
can also create high-quality orthoradial or hexalinear maps. The main limitation of our
approach so far is that station labels are not yet considered during the schematization.
Although we achieved satisfying results by using a simple a posteriori labeling approach,
we emphasize that further research is required here. It seems promising to reconsider
the labeling approach from an early metro map drawing approach by Stott et al. [134]
and apply it to our work. Another practically relevant problem that has received rela-
tively little research interest so far is the automatic drawing of schematic transit maps
with tari zones. Here a promising approach seems to be to use the edges of the zone
polygons as special edges in the input line graph.
Finally, we gave a practically oriented overview of how the free line graphs, either
schematic or geographically accurate, and with optimal line permutations, can be ren-
dered into maps in an esthetically pleasing and informative way. This is in itself a fas-
cinating topic of which we have merely scratched the surface, and there is recent and
ongoing research to quantify the esthetic quality of metro maps. We consider it highly
promising to use the approach described in this work to conduct a study on perceived
map quality. For example, we may render transit maps using dierent layouts, dier-
ent line orderings, and dierent styles for stations and segments, and do A/B testing on
them.
We are also convinced that the software pipeline developed in the course of this work
can serve as a solid framework for future research in the area of transit map drawing.
In particular, all our tools are designed in such a way that they can be easily substituted
by drop-in replacements.
260 Chapter 8. Conclusions
List of Figures
1.1 London subway map in 1908 and 1931. Images taken from [92], © Transport
for London (TfL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Chicago loop in Google Maps, Here Maps, and the ocial CTA map.
© Chicago Transit Authority . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Excerpt from the ocial 2019 Swiss timetable map. © Schweizerische
Bundesbahnen AG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Excerpt from the ocial 2020 regional rail map of southwestern Ger-
many. © Deutsche Bahn AG . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 Excerpt from a line graph for the Chicago light rail network . . . . . . . 8
2.1 Map-matching a train trip through Switzerland. Figure adapted from [14] 16
2.2 Transit routes in Google Maps, the DB journey planner, and Bing maps . 17
2.3 Correct and erroneous shapes in GTFS data . . . . . . . . . . . . . . . . . 18
2.4 Problems with transition only between adjacent road segments . . . . . 22
2.5 Transportation network N with a map-matching problem . . . . . . . . 28
2.6 Distance distribution between similar station pairs in OSM. Figure taken
from [15] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.7 HMM for map-matching a 3 stop trip . . . . . . . . . . . . . . . . . . . . 30
2.8 Optimal hidden state sequence as a shortest path problem . . . . . . . . 33
2.9 Modeling turn restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.10 Angular turn costs. Figure taken from [14] . . . . . . . . . . . . . . . . . 35
2.11 Turn costs between HMM transitions. Figure taken from [14] . . . . . . 36
2.12 Paths in primal and edge-to-node dual graphs . . . . . . . . . . . . . . . 36
2.13 Encoding vehicle orientations in hidden states. Figure taken from [14] . 38
2.14 Shortest paths in primal and edge-to-node dual graphs . . . . . . . . . . 40
2.15 Shortest paths to nodes without outgoing edges in the dual graph . . . . 40
2.16 Turn restrictions in edge-to-node duals . . . . . . . . . . . . . . . . . . . 41
2.17 Labeling variants for Freiburg Main Station. Figure taken from [15] . . . 43
261
262 Chapter 8. List of Figures
2.18 Station identiers for London St Pancras in 3 dierent datasets. Figure
taken from [15] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.19 Interwoven grids to encode approximate regions. Figure taken from [15] 52
2.20 Single-Dijkstra calculation of transition probabilities . . . . . . . . . . . 53
2.21 Trip trie and corresponding HMM instance graph . . . . . . . . . . . . . 58
2.22 FN and FP for a Jaccard index based classier. Figure taken from [15] . . 62
2.23 FN and FP for a PED based classier. Figure taken from [15] . . . . . . . 62
2.24 FN and FP for a TFIDF based classier. Figure taken from [15] . . . . . . 63
2.25 FN and FP for a random forest based classier. Figure taken from [15] . . 63
2.26 A
10
N
scores for DIST-DIFF and OURS under Gaussian noise . . . . . . . . 67
2.27 Considering scheduled travel times during map matching . . . . . . . . . 73
3.1 Line graph and its rendering, prepared with a naive approach . . . . . . 76
3.2 Line graph and its rendering, prepared with a constructive approach . . 76
3.3 Constructing the support graph H
j
for an input graph G . . . . . . . . . 81
3.4 Two types of artifacts left by a single iteration of our approach . . . . . . 83
3.5 Line creep issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.6 No-merge zones for sampling points . . . . . . . . . . . . . . . . . . . . . 84
3.7 Artifacts in nonplanar networks . . . . . . . . . . . . . . . . . . . . . . . 85
3.8 Avoiding artifacts using no-merge zones . . . . . . . . . . . . . . . . . . 85
3.9 Intersection smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.10 Chicago loop, with and without line turn restrictions considered . . . . . 87
3.11 Turn restrictions distorted during the support graph construction . . . . 88
3.12 Adding turn restrictions onto a support graph H . . . . . . . . . . . . . . 89
3.13 Lines not serving the clustered Bellevue station node in Zurich . . . . . . 93
3.14 Examples of remaining line graph construction problems . . . . . . . . . 94
3.15 Line graph for the 2022 tram network of Freiburg . . . . . . . . . . . . . 97
3.16 Line graph for the 2020 light rail network of Chicago . . . . . . . . . . . 98
3.17 Excerpt from the line graph for the 2018 subway network of New York . 99
3.18 Excerpt from the line graph for the 2021 tram network of Zurich . . . . . 100
3.19 Line graph of the 2021 Zurich tram network around the city center . . . 101
4.1 Chicago Loop with and without optimized line ordering . . . . . . . . . 104
4.2 Line orderings in MLNCM and MLCM . . . . . . . . . . . . . . . . . . . 105
4.3 Line crossing types in transit maps . . . . . . . . . . . . . . . . . . . . . 106
4.4 Line orderings for MLNCM and MLCM . . . . . . . . . . . . . . . . . . . 107
263
4.5 Periphery condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.6 Line orderings with and without considering line separations . . . . . . . 112
4.7 Line separations, detail . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.8 Periphery condition enforced by line separation penalty . . . . . . . . . 113
4.9 Periphery condition violated in separation-free ordering . . . . . . . . . 113
4.10 Optimal line ordering with only line separations considered . . . . . . . 114
4.11 Clustering crossings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.12 Solving MLCM-P with an MLNCM gadget . . . . . . . . . . . . . . . . . 116
4.13 Counting crossings and separations . . . . . . . . . . . . . . . . . . . . . 118
4.14 Illustration of the proof for Theorem 4.9 . . . . . . . . . . . . . . . . . . 121
4.15 Crossing oracle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.16 Line combination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4.17 Moving crossings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4.18 Illustration of Lemma 4.13 . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4.19 Illustration of Lemma 4.15 . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4.20 Illustration of Pruning Rule 1 (Node Contraction). Figure taken from [20] 133
4.21 Illustration of Pruning Rule 2 (Line Partner Collapse). Figure taken from
[20] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.22 Illustration of Pruning Rule 3 (Single Edge Prune). Figure taken from [20] 134
4.23 Illustration of Cutting Rule 1 (Single Line Cut). Figure taken from [20] . 135
4.24 Illustration of Cutting Rule 2 (Terminus Detachment). Figure taken from
[20] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.25 Pruning & cutting a line graph. Figure taken from [19] . . . . . . . . . . 136
4.26 Full X in the New York subway network. Figure taken from [20] . . . . . 136
4.27 Double-Y in the Stuttgart light rail network. Figure taken from [20] . . . 137
4.28 Illustration of Untangling Rule 1. Figure taken from [20] . . . . . . . . . 137
4.29 Illustration of Untangling Rule 2. Figure taken from [20] . . . . . . . . . 138
4.30 Illustration of Untangling Rule 3. Figure adapted from [20] . . . . . . . . 139
4.31 Illustration of Untangling Rule 4, with inverse minor legs. Figure taken
from [20] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
4.32 Illustration of Untangling Rule 4, with matching minor legs. Figure taken
from [20] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
4.33 Illustration of Untangling Rule 5. Figure adapted from [20] . . . . . . . . 142
4.34 Illustration of Untangling Rule 6 . . . . . . . . . . . . . . . . . . . . . . . 143
4.35 Illustration of Untangling Rule 7 . . . . . . . . . . . . . . . . . . . . . . . 144
4.36 Naive application of a single round of Pruning Rule 1 (Node Contraction) 145
264 Chapter 8. List of Figures
4.37 Illustration of the proof for Lemma 4.38 . . . . . . . . . . . . . . . . . . . 148
4.38 Untangling opportunities created by an untangling rules. . . . . . . . . . 151
4.2 Full pruning, cutting and untangling of a line graph . . . . . . . . . . . . 152
4.39 Full solve of a tree-like line graph through simplication . . . . . . . . . 158
4.40 Simplifying a two-sided tree-like line graph . . . . . . . . . . . . . . . . . 159
4.41 Optimal line ordering for Stuttgart via line graph simplication . . . . . 161
4.42 Chicago Loop line ordering in the ocial map (© Chicago Transit Au-
thority), and using our approach . . . . . . . . . . . . . . . . . . . . . . . 172
4.43 Generic ripping rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
4.44 Induced untangling rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
4.45 Line-ordering optimized map of the New York subway . . . . . . . . . . 176
5.1 Degree-2 heuristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
5.2 Extending the maximum input node degree. Figure taken from [22] . . . 179
5.3 Example results of previous work on schematic maps. Figures taken from
[12, 33, 144, 101] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
5.4 Example results of previous work on octilinear transit maps. Figures
taken from [84, 106, 134, 118, 142, 38, 143, 21] . . . . . . . . . . . . . . . . 182
5.5 Example results of previous work on non-octilinear schematic maps. Fig-
ures taken from [58, 48, 112, 113] . . . . . . . . . . . . . . . . . . . . . . 185
5.6 Shortest paths through grid graphs with and without bend penalties. Fig-
ure taken from [21] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
5.7 An octilinear grid graph and its extended version. Figure taken from [21] 189
5.8 Shortcuts in traversing an extended grid node. Figure taken from [21] . . 190
5.9 A path through a grid graph and its extended version. Figure taken from
[21] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
5.10 Metro-Map image solving the Hamilton path problem . . . . . . . . . . . 194
5.11 Edge processing order and corresponding drawing. Figure taken from [21] 200
5.12 Finding the image for an input edge. Figure taken from [21] . . . . . . . 202
5.13 Blocking sink edges to avoid stalling. Figure adapted from [21] . . . . . . 203
5.14 Sink edge weights reecting bend costs. Figure adapted from [21] . . . . 204
5.15 Blocking sink edges to preserve circular ordering. Figure adapted from
[21] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
5.16 Free node degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
5.17 Three steps during our local search. Figure taken from [21] . . . . . . . . 206
5.18 Spring embedding enforcing uniform distance between re-inserted nodes.
Figure taken from [21] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
265
5.19 Cases to consider for an octilinear grid heuristic . . . . . . . . . . . . . . 208
5.20 Convex hull based sparse grid. Figure taken from [22] . . . . . . . . . . . 210
5.21 Quadtree based sparse grid. Figure taken from [22] . . . . . . . . . . . . 211
5.22 Octilinear Hanan grid based sparse grid. Figure taken from [22] . . . . . 212
5.23 Hanan grid iterations. Figure taken from [22] . . . . . . . . . . . . . . . . 213
5.24 Ocial octilinear long-distance network map of the Swiss Federal Rail-
ways. © Schweizerische Bundesbahnen AG . . . . . . . . . . . . . . . . . 214
5.25 Respecting obstacles in octilinear maps . . . . . . . . . . . . . . . . . . . 214
5.26 Constraint violations in a metro-map image. Figure taken from [22] . . . 215
5.27 Hexalinear grid graph. Figure taken from [22] . . . . . . . . . . . . . . . 216
5.28 Orthoradial and pseudo-orthoradial base grids. Figure taken from [22] . 217
5.29 Freiburg tram network on dierent sparse grids. Figure taken from [22] . 225
5.30 Fish-eye eect and enlarged cut out in the Chicago transit map . . . . . . 229
5.31 Input line graph rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
5.32 Octilinear map of the Vienna subway network . . . . . . . . . . . . . . . 232
5.33 Orthoradial map of the Vienna subway network . . . . . . . . . . . . . . 232
5.34 Octilinear map of the Freiburg tram network . . . . . . . . . . . . . . . . 233
5.35 Orthoradial map of the Freiburg tram network . . . . . . . . . . . . . . . 233
5.36 Octilinear map of the Berlin subway network . . . . . . . . . . . . . . . . 234
5.37 Orthoradial map of the Berlin subway network . . . . . . . . . . . . . . . 234
5.38 Octilinear map of the Sydney light rail network . . . . . . . . . . . . . . 235
5.39 Orthoradial map of the Sydney light rail network . . . . . . . . . . . . . 235
5.40 Octilinear map of the Stuttgart light rail network . . . . . . . . . . . . . 236
5.41 Orthoradial map of the Stuttgart light rail network . . . . . . . . . . . . 237
5.42 Hexalinear map of the Vienna subway network . . . . . . . . . . . . . . 238
5.43 Hexalinear map of the Berlin subway network . . . . . . . . . . . . . . . 238
5.44 Octilinear map of the London subway network . . . . . . . . . . . . . . . 239
5.45 Octilinear map approximating geographical line courses . . . . . . . . . 240
5.46 Ortholinear map respecting obstacles . . . . . . . . . . . . . . . . . . . . 240
6.1 Overview over our rendering approach . . . . . . . . . . . . . . . . . . . 242
6.2 Osetting a polyline by a distance d . . . . . . . . . . . . . . . . . . . . . 243
6.3 Clipping of oset spikes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
6.4 Self-intersections in polyline osetting . . . . . . . . . . . . . . . . . . . 244
6.5 A cubic Bézier curve B . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
6.6 Rendering inner node line connections using cubic Bézier curves . . . . 245
266 Chapter 8. List of Figures
6.7 Dierent station marker styles in manually designed maps . . . . . . . . 246
6.8 Freiburg tram network with and without station markers . . . . . . . . . 246
6.9 Station rendering using the convex hull of the node fronts . . . . . . . . 247
6.10 Other methods of station rendering . . . . . . . . . . . . . . . . . . . . . 247
6.11 Methods of station rendering for unserved lines . . . . . . . . . . . . . . 247
6.12 Excerpt from the labeled Stuttgart light rail network map . . . . . . . . . 248
6.13 Station labeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
6.14 Labeled geographically accurate map of the Stuttgart light rail network . 250
7.1 Station similarity classication GUI for statsimi . . . . . . . . . . . . . . 252
7.2 Our web tool staty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
7.3 Rendering of a simple input line graph by our tool transitmap . . . . . . 255
7.4 Curvilinear transit map of Stuttgart using a drop-in replacement for octi 256
List of Tables
2.1 Dimensions and shape quality of GTFS feeds . . . . . . . . . . . . . . . . 19
2.2 Example feature vectors for three station pairs . . . . . . . . . . . . . . . 51
2.3 Dimensions of ground-truth datasets BI and DACH . . . . . . . . . . . . 60
2.4 Station similarity classication evaluation results . . . . . . . . . . . . . 61
2.5 Dimensions of our map-matching testing datasets . . . . . . . . . . . . . 65
2.6 A
10
N
scores without noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.7 δ
aF
scores without noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.8 A
10
N
scores under Gaussian noise . . . . . . . . . . . . . . . . . . . . . . . 70
2.9 δ
aF
scores under Gaussian noise . . . . . . . . . . . . . . . . . . . . . . . 70
2.10 Running times of our approach using several speed-up techniques . . . . 71
3.1 Quality of our line graph construction approach . . . . . . . . . . . . . . 92
3.2 Time and iterations required for the line graph construction . . . . . . . 95
4.1 Known complexities of MLCM and MLNCM subproblems . . . . . . . . . 110
4.2 Asymptotic complexities of single rounds of simplication rules . . . . . 157
4.3 Line graphs used in our experimental evaluation . . . . . . . . . . . . . . 162
4.4 Main results for the running times . . . . . . . . . . . . . . . . . . . . . . 163
4.5 Main results for the relative approximation errors . . . . . . . . . . . . . 164
4.6 Comparison of the quality and speed of our approximate approaches . . 165
4.7 Average solution times and approx. errors of heuristic approaches . . . . 166
4.8 ILP Dimensions and solution times on the raw and on the pruned & cut
input graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
4.9 Eects of full line graph simplication on line graph dimensions . . . . . 169
4.10 Impact of full simplication on ILP sizes and solution times . . . . . . . . 170
4.11 Impact of full simplication on selected baseline heuristics . . . . . . . . 171
4.12 Comparison of our found line orderings to manually designed maps . . . 173
267
268 Chapter 8. List of Tables
5.1 Reported octilinearization times from previous work . . . . . . . . . . . 184
5.2 Line graphs used in our experimental evaluation . . . . . . . . . . . . . . 218
5.3 ILP dimensions, solution times, and objective values for LP-2 . . . . . . . 220
5.4 Comparison of the performance of dierent ILP solvers . . . . . . . . . . 221
5.5 Eects of input edge ordering methods on approximation quality . . . . 222
5.6 Quality of our approximate approach with deg-2 heuristic (A-2) . . . . . 223
5.7 Solution times for our ILP and our approximate approach . . . . . . . . . 223
5.8 Eects of sparse base grids on grid graph size . . . . . . . . . . . . . . . 224
5.9 Eects of base grid simplication on ILP sizes . . . . . . . . . . . . . . . 226
5.10 Eects of sparse base grids on our approximate approach . . . . . . . . . 227
5.11 Memory consumption on sparse base grids . . . . . . . . . . . . . . . . . 227
5.12 Solution time and approximation errors of alternative layouts . . . . . . 228
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